Robert A. Crovelli Education: BS in Mechanical Engineering at Bucknell University MS in Mathematics at Bucknell University MS in Probability at Michigan State University PhD in Statistics at Colorado State University Experience: Professor of probability and statistics for 15 years at Bucknell University, Lycoming College, and Metropolitan State College of Denver Mathematical statistician at United States Geological Survey for 25 years Presently retired Publications PhD thesis: Stochastic Models for Precipitation Textbook: Principles of Statistics and Probability USGS manual: Probability and Statistics for Petroleum Resource Assessment Approximately 100 papers; most papers in probabilistic oil and gas resource assessment Last paper: Crovelli, R.A. and Coe, J.A., 2009, Probabilistic estimation of numbers and costs of future landslides in the San Francisco Bay region, Georisk, Vol. 3, No. 4, 206223. ABSTRACT Many basic concepts in applied mathematics are incomplete or incorrect. The purpose of this analysis is to complete and correct many basic concepts in applied mathematics. The objective of the study is to shore up the basic concepts in order to solidify the foundation of applied mathematics. The topics discussed are grouped into two parts with their sections as follows: Part I. Numbers and their relations: sets of objects, sets of numbers, kinds of numbers, operations of numbers, and properties of operations; Part II. Variables and their relations: constants, variables, relations, probability, and statistics. The basic concepts are related as follows: numbers depend on sets, variables depend on numbers, probability depends on variables, and statistics depends on probability. A new theory of various kinds of numbers is developed with respect to how numbers are used in the world. The theory fundamentally enhances our understanding of the meaning of numbers and all of the basic concepts that depend on numbers. This analysis carefully establishes the basic concepts that form the foundation of applied mathematics which is used in applications of science, engineering, and economics.

AN ANALYSIS OF THE BASIC CONCEPTS IN APPLIED MATHEMATICS ROBERT A. CROVELLI INTRODUCTION Many basic concepts in applied mathematics are incomplete or incorrect. The purpose of this analysis is to complete and correct many basic concepts in applied mathematics. The objective of the study is to shore up the basic concepts in order to solidify the foundation of applied mathematics. The topics discussed are grouped into two parts with their sections as follows: Part I. Numbers and their relations: sets of objects, sets of numbers, kinds of numbers, operations of numbers, and properties of operations; Part II. Variables and their relations: constants, variables, relations, probability, and statistics. The basic concepts are related as follows: numbers depend on sets, variables depend on numbers, probability depends on variables, and statistics depends on probability. A new theory of various kinds of numbers is developed with respect to how numbers are used in the world. The theory fundamentally enhances our understanding of the meaning of numbers and all of the basic concepts that depend on numbers. This analysis carefully establishes the basic concepts that form the foundation of applied mathematics which is used in applications of science, engineering, and economics. PART I: NUMBERS AND THEIR RELATIONS Commentary on The Mysteries of Arithmetic by James R. Newman in The World of Mathematics (1956): “Arithmetic is commonly supposed to be the simplest branch of mathematics. Nothing could be further from the truth. The subject is difficult from the ground up, though the practice of elementary arithmetic is admittedly easy enough. The same can be said of most sciences: it is unnecessary to understand electromagnetic theory before wiring a lamp or to study physics in order to repair a pump. We count on our fingers and give no heed to the proliferating implications of the act. The fundamental rules and operations of arithmetic are extraordinarily hard to define.” Mathematics is the science of numerical relationships. The two main divisions of mathematics are pure and applied mathematics. Pure mathematics is the science of abstract-numerical relationships. Applied mathematics is the science of physical-numerical relationships.

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Robert A. Crovelli holds a B.S. in mechanical engineering from Bucknell University and a Ph.D. in mathematical statistics from Colorado State University. He is currently retired from the United States Geological Survey. This paper is the detailed version of a seminar given at the Army Corps of Engineers in Lakewood, Colorado, on 7 March, 2011.

SETS OF OBJECTS A set is a collection of objects or things, called elements or members of the set. Sets are usually denoted by A, B, and C. Ex. Football team, family of people, set of golf clubs, herd of elk, gaggle of geese, set of ideas, list of activities, bouquet of flowers Ex. Sets that are pairs of objects: husband and wife, pair of gloves, pair of shoes A general set consisting of a pair of things is denoted by {α, β}. Subset Set B is a subset of set A, denoted B ⊂ A, if every element of B is also an element of A. Ex. All of the subsets of A = {α, β}: { } {α} {β} {α, β} where {} denotes the empty set. Set operations: Union Set A union set B, denoted A ∪ B or AorB, is the set of all those elements in either A or B, or both. Ex. Given A = {α, β} and B = {β}, then A ∪ B = {α, β} Intersection Set A intersection set B, denoted A ∩ B or AandB, is the set of all those elements in common to A and B. Ex. Given A = {α, β} and B = {β}, then A ∩ B = {β} Disjoint When the intersection of two sets is the empty set, the sets are said to be disjoint or mutually exclusive sets. Ex. Given A = {α} and B = {β}, then A ∩ B = {} so A and B are disjoint sets. Separation Set A separation subset B, denoted A ~ B or AnotB, is the set of all those elements in A not in B. Ex. Given A = {α, β} and B = {β}, then A ~ B = {α} Ex. Given A = U universal set and subset B, then U ~ B = B’ complement of B SETS OF NUMBERS A digit is a symbol. The set of digits D is given by D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} A digit is like a letter of the alphabet; a word is a string of one or more letters. Ex. dog. A number is a symbol formed by a string of one or more digits. Ex. 251. Important sets of numbers: natural or counting, whole, integers, rational, irrational, and real. The set of natural numbers or counting numbers N is given by N = {1, 2, 3, 4, 5, …} The set of whole numbers W is given by W = {0, 1, 2, 3, 4, 5, …}

The set of integers Z is given by Z = {…, -3, -2, -1, 0, 1, 2, 3, …} The set of rational numbers P is given by P = {a/b | a  Z, b  Z, and b ≠ 0} Rational numbers have decimal representations which are either terminating or nonterminating repeating. The set of irrational numbers Q is given by Q = decimals which are nonterminating nonrepeating. The set of real numbers R is given by R=PUQ There is a one-to-one correspondence between real numbers and points on a number line. Every point on the number line represents a real number. An ordinary ruler is very much like a physical version of a positive number line. Number line: . | | | | | | | . … -3 -2 -1 0 1 2 3 … There are also imaginary numbers I, complex numbers C, and transfinite numbers T. The sets of numbers are nested: N ⊂ W ⊂ Z ⊂ P ⊂ R ⊂ C. KINDS OF NUMBERS At this stage of development of numbers, we have established that numbers are simply symbols which have a one-to-one correspondence with points on the number line. This knowledge is sufficient for a purely mathematical treatment of numbers, and the standard approach is to proceed onto numerical operations and their properties. Pure mathematics is not concerned with relations of numbers to the physical world. Applied mathematics is concerned with relations of numbers to the physical world. An extremely important applied mathematical issue is the meaning of numbers as they relate to the physical world. The material in this section is a new theory of the various kinds of numbers with respect to how numbers are used in the world. The objective is to establish their meaning on a firm or solid foundation. We use numbers but do not completely understand their nature. We use senses but do not completely understand their nature. By studying the number line, several basic properties of numbers can be observed. Basic properties of numbers (for simplicity consider just the whole numbers): Distinctive property: The numbers consist of different symbols. Ordinal property: The numbers are ordered in that they follow one another in steps from left to right. Quantitative property:

The larger the distance a number is from zero, the larger the quantity associated with that number; also, the larger the symbol for the number in terms of more digits. The order of presentation of these three properties of numbers is important. They are presented such that their corresponding concepts are increasing in complexity. The concept of order is more complex than the concept of distinction. The concept of quantity is more complex than the concept of order. The three related concepts can be thought of as building on one another successively. Letters of the alphabet and words, of course, possess the distinctive property (e.g., names of people) and the ordinal property to some extent (e.g., A, B, and C), but not the quantitative property except for some descriptive terms like “small,” “medium,” and “large” which are quite gross compared to the high precision of the real numbers or the number line. Set-number functions are used to relate or establish a correspondence between things and numbers, so that numbers are directly associated with things. Functional notation is used to express the set-number functions (relations). A set-number function is a correspondence (or rule) between a set of objects and a number or a set of numbers. Each kind of number has its own set-number function. A set-number function will be defined for each kind of number. A classification scheme is established for how numbers are used in the world. There are four kinds of numbers with respect to how numbers are used in the world. Mathematical definitions of the four kinds of numbers are established in terms of setnumber functions. Four kinds of numbers: identification, ordinal, and cardinal (count and measure). Identification numbers are numbers used to represent or describe “distinction.” Ordinal numbers are numbers used to represent or describe “position.” Count numbers are numbers used to represent or describe “how many.” Measure numbers are numbers used to represent or describe “how much.” Cardinal numbers are numbers used to represent or describe “quantity.” Science and engineering use (1) measure, (2) count, (3) ordinal, (4) identification numbers in order of importance. The establishment of the various kinds of numbers involves defining relationships between numbers and sets of things in the world. Four kinds of set-number functions will be defined: Identification functions relate sets of objects to identification numbers. Ordinal functions relate sets of objects to ordinal numbers. Count functions relate sets of objects to count numbers. Measure functions relate sets of objects to measure numbers. Identification numbers

Identification numbers utilize the distinctive property of numbers. An identification number is a number that represents (or relates to) a code for identifying or distinguishing between things. The set of all identification numbers is equal to the set of whole numbers W. Ex. Sports shirt number, social security number, driver’s license number, birth date, bank account number, credit card number, train number, telephone number, catalog number Ex. Great sports figures are honored by having their jersey number retired. Qualitative data can be coded by identification numbers to make the data numerical. Ex. The answer to a yes-or-no question can be coded by yes ≡ 1 and no ≡ 0. Identification functions relate sets of objects to identification numbers. Given any set of the form {α} where α represents an object. Identification function is defined: I({α}) = {a}where a is a whole number. Given any set of the form {α, β} where α and β represent a pair of objects. Identification function is defined: I({α, β}) = {a, b}where a and b are whole numbers and a ≠ b. Given any set of the form {α1, α2, …, αn} where αi represents an object; i = 1, 2, …, n Identification function is defined: I({α1, α2, …, αn}) = {a1, a2, …, an}where ai is a whole number and ai ≠ aj; i = 1, 2, …, n and j = 1, 2, …, n. Special case: 1 ≤ ai ≤ n which is a subset of W In this special case the total number of possible assignments is n! permutations. Ex. Given a new football team of 50 players. Identification numbers on their jerseys can range from 1 to 50. The total number of possible assignments of jersey numbers is 50! = 50*49*…*2*1 Suppose the coach of the football team makes a table with the following four column headings (and 50 rows) to assist in assigning the jersey numbers: 1) Name of player; 2) Salary of player; 3) Rank of player according to salary with the highest salaried player having a rank of 1; and 4) Jersey number of player. After the first three columns are filled in, the player with rank 1 can choose any one of the 50 available jersey numbers. The selected number is written in the fourth column of the table. The player with rank 2 can choose any one of the 49 remaining available jersey numbers; the selected number is written in the fourth column of the table. This procedure continues until all of the jersey numbers have been selected. This problem involves three kinds of numbers: identification, ordinal, and cardinal numbers: salary (amount of money) is a cardinal number; rank (order of selection) is an ordinal number; and jersey number (identification of player) is an identification number. Ordinal numbers Ordinal numbers utilize the ordinal property of numbers.

An ordinal number is a number that represents (or relates to) an order in a series. The set of all ordinal numbers is equal to the set of natural numbers N. The ordinal number of a discrete set is the number used to describe the “position” of an element in that set which is arranged in order. Ex. Series of books: first (1) book, second (2) book, third (3) book, … Ex. Page number, chapter number, birth order, today’s date, step number, list number, instruction number, lesson number, baseball inning number, race finishing number. Ordinal numbers are similar to how letters A, B, C, … can be used to show order. Ordinal functions relate sets of objects to ordinal numbers. Given any set of the form {α} where α represents an object. Ordinal function is defined: O({α}) = {1} Given any set of the form {α, β} where α and β represent a pair of objects. Ordinal function is defined: O({α, β}) = {1, 2} Given any set of the form {α1, α2, …, αn} where αi represents an object; i = 1, 2, …, n Ordinal function is defined: O({α1, α2, …, αn}) = {1, 2, …, n} Cardinal numbers “Mathematics, or the science of magnitudes, is that system which studies the quantitative relations between things” (P.D. Ouspensky, 1970). Cardinal numbers utilize the quantitative property of numbers. A cardinal number is a number that represents (or relates to) a quantity. Cardinal numbers are associated with quantities and used to describe and compare quantities. Quantitative relationships are established between things and numbers. Cardinal numbers are conceptual entities. The cardinal number 2 which represents a quantity is a concept. An identification number is a “specific” concept. Ex. A social security number identifies a specific person, like a name. An ordinal number and a cardinal number are both “general” concepts. Ex. “Second” is a general position; “two” is a general quantity. Practically all of mathematics is concerned with cardinal numbers. Mathematics is essentially the science or study of quantitative relationships. Two kinds of cardinal numbers: count and measure numbers. Approach to development of cardinal numbers: Sets of objects are associated with sets of numbers which are related to cardinal numbers.

Discrete sets of objects are associated with discrete sets of numbers which are related to count numbers. Continuous sets of objects are associated with continuous sets of numbers which are related to measure numbers. Note: Discrete ≡ digital; continuous ≡ analog. Count numbers A count number is a number that represents (or relates to) a discrete quantity, describing how many. The set of all count numbers is equal to the set of integers Z. The count number of a discrete set is the number used to describe “how many” elements there are in that set. The count number 2 represents the quantity of (or how many) things are in any set consisting of a pair of things. Ex. Sets of books: one (1) book, two (2) books, three (3) books, … Ex. Number of people, number of cars, number of trips, number of earthquakes, number of oil wells, amount of money, number of ideas, number of facts, number of reasons. Count functions relate sets of objects to count numbers. Given any set of the form {α} where α represents an object. Count function is defined: C({α}) = 1 C({ }) = 0 Given any set of the form {α, β} where α and β represent any objects. Count function is defined: C({α, β}) = 2 set has a certain form of “twoness” On the other hand, recall that O({α, β}) = {1, 2} C({ordinal number 1, ordinal number 2}) = 2 count or cardinal number Association and assignment of ordinal numbers: Discrete sets of objects are associated with discrete sets of numbers (ordinals) which are related to count numbers. We assign ordinal numbers using any systematic order to the members in the set of objects for the purpose of establishing the last ordinal number in the ordinal set whose value is used to determine the count or cardinal number of the set (cardinality of the set). Given any set of the form {α1, α2, …, αn} where αi represents an object; i = 1, 2, …, n Count function is defined: C({α1, α2, …, αn}) = n On the other hand, recall that O({α1, α2, …, αn}) = {1, 2, …, n} C({1, 2, …, ordinal number n}) = n count or cardinal number Definition of a count number in terms of a count function: C({1, 2, …, n}) = n count number

The count number of the discrete set of natural numbers from 1 to the last number n (an ordinal number), denoted by {1, 2, …, n}, is numerical equal to the value of the last number n. The count number n of the discrete set {1, 2, …, ordinal number n} is equal to the value of the last ordinal number n in the set. The count function is an integer-valued set function. This set function defines a relation between ordinal and count numbers. The set function describes the process of ordinary counting that is used to determine a count number. In the counting method ordinal numbers are used to determine a count number. The counting method for determining count numbers uses the counting numbers N. The associating method The associating method for determining the count number of a set is when the count function in the form C({α1, α2, …, αn}) = n is used by associating the set in question directly with a count number. The associating method is applicable where the count number is relatively small, say at most around 5. Ex. When pharmacists count out pills, they usually separate out at most 5 pills at a time. Ex. Application of the associating method. A woman goes to a park and notices a gaggle of geese. She observes immediately that the number of geese is 4. Her observation of 4 geese was done by associating and not counting. The reason for using the associating method was that the number of geese was relatively small. The woman over time had experienced numerous groups of 4 objects and was able to simply associate the count number 4 to such groups without counting one by one. The number of geese was not large enough to require counting. If the number of geese had been 14 instead of 4, then she would have had to use the counting method to determine the number of geese. The counting method The counting method for determining the count number of a set is when the count function in the form C({1, 2, …, ordinal number n}) = n is used by counting the members in the set in question to establish the ordinal number n of the last member of the set; the count or cardinal number of the set is equal in value to n. The counting method is usually applied only when the count number is relatively large, say more than around 5. Algorithm of the counting method for determining the count number of a discrete set of things: 1. Associate the discrete set of natural numbers {1, 2, …, n} with the discrete set of things. 2. Establish the value of the last number n (an ordinal number) in the discrete set of natural numbers {1, 2, …, n}. 3. The count number of the discrete set of natural numbers {1, 2, …, n} is numerical equal to the value of the last number n (an ordinal number). (This is the definition of a count number.)

4. The count number of the discrete set of things is equal to the count number of the discrete set of natural numbers {1, 2, …, n} because of the association. Note that a numerical set is associated with a physical set, and the cardinal number of the numerical set is used to determine the cardinal number of the physical set. Ex. Application of the counting method. Given a classroom full of students, we want to know how many students are in the classroom. Using the counting method, we start counting the students one by one assigning 1 to the first student at the end of the first row, 2 to the second student who is next to the first student, and continue in this manner through all of the rows until we get to the last student in the last row. Suppose the last student is assigned the ordinal number 30 being the thirtieth student in the classroom. Then we can state that the count number of students in the classroom is 30. Note that the counting pattern needs to be systematic in order to keep tract of who has already been counted and who is yet to be counted. The order itself is not important. Comparison of identification, ordinal, and count numbers Given any set of the form {α, β} where α and β represent a pair of objects. Identification function is defined: I({α, β}) = {a, b}where a and b are whole numbers and a ≠ b. Ordinal function is defined: O({α, β}) = {1, 2} Count function is defined: C({α, β}) = 2 set has a certain form of “twoness” Relationship between consecutive numbers (say n and n + 1 where n is a natural number) within each of the three kinds of numbers: Identification numbers: n + 1 is different from n Ordinal numbers: n + 1 is next position after n Count numbers: n + 1 is one more than n Count numbers are used to compare discrete quantities. 1 is less than 2, denoted as 1 < 2, because of the following two reasons: 1) The general set {α} has one less object than the general set {α, β}, i.e., {α} < {α, β}. 2) The distance on the number line that 1 is from 0 is less than the distance on the number line that 2 is from 0. The reason is not simply because 1 is to the left of 2 on the number line which is an ordinal relation. Usually in mathematics books the relation 1 < 2 is given or assumed or defined and not explained why it is so. In pure mathematics the relation 1 < 2 has no physical meaning. Measure numbers A measure number is a number that represents (or relates to) a continuous quantity, describing how much. The set of all measure numbers is equal to the set of real numbers R.

The measure number of a continuous set is the number used to describe “how much” quantity there is in that set. Ex. distance (including length, width, height, depth), area, volume, weight, time (age), temperature, speed, density, porosity, permeability, viscosity, loudness of sound. A measure number gets its measure of quantity by its distance from zero. The larger the distance from zero, the larger the measure of quantity. A line segment is equivalent to an interval; a distance is equivalent to a length. A distance or a length is a function of a line segment or an interval. The interval from a to b, denoted [a, b], is the set of all real numbers (points) between a and b inclusive. Using set notation, we have [a, b] ≡ {r | r is a real number and a ≤ r ≤ b} The interval from 0 to a, denoted [0, a], is the set of all real numbers (points) between 0 and a inclusive. Using set notation, we have [0, a] ≡ {r | r is a real number and 0 ≤ r ≤ a} Association with intervals: Continuous sets of objects are associated with continuous sets of numbers (intervals of the form [0, a]) which are related to measure numbers. The value of the last number or end-point a in the interval [0, a] is used to determine the measure or cardinal number of the set (cardinality of the set). The last number or end-point a in the interval [0, a] can be thought of as an “ordinal” number because what is important about this number is its position in the set. Then the ideas in the development of the concept of a measure number would exactly parallel the ideas in the development of the concept of a count number. The only difference would be that measure numbers are concerned with continuous sets while count numbers are concerned with discrete sets. Measure functions relate sets of objects (using intervals) to measure numbers. Given any interval of the form [0, a]. Measure function is defined: M([0, 0]) = 0 M([0, 1]) = 1 M([0, 2]) = 2 M([0, n]) = n measure number where n is a real number and n ≥ 0. Definition of a measure number in terms of a measure function: M([0, n]) = n measure number The measure number (a length) of the continuous set of real numbers from 0 to the last number or end-point n (an ordinal number), denoted by [0, n] (an interval), is numerical equal to the value of the last number or end-point n. The measure number of the continuous set [0, n] is equal to the value of the end-point or last number in the set. The measure number represents the length of the interval [0, n] or distance from zero.

End-point of an interval (ordinal number) versus length of an interval (measure number). The measure function is a real-valued set function. Algorithm of the measuring method for determining the measure number of a continuous set of interest: 1. Associate the continuous set of real numbers [0, n] (an interval) with the continuous set of interest. 2. Establish the value of the last number or end-point n (an ordinal number) in the continuous set of real numbers [0, n] (an interval). Whenever any measurement is made in practice, the end-point n of the interval [0, n] is established. 3. The measure number (a length) of the continuous set of real numbers [0, n] (an interval) is numerical equal to the value of the last number or end-point n (an ordinal number). (This is the definition of a measure number.) 4. The measure number of the continuous set of interest is equal to the measure number of the continuous set of real numbers [0, n] because of the association. Note that a numerical set is associated with a physical set, and the cardinal number of the numerical set is used to determine the cardinal number of the physical set. A very important relationship can be established between the count and measure functions: C({1, 2, … , ordinal number n}) = n and M([0, n]) = n In both cases the cardinal number of the set is equal in value to the last number in the set. Realize that this is an important relationship between two important relations. In actual practice the last number or end-point in the set is determined (counted or measured), and its value is used to establish the cardinal number of the set. The last number or end-point itself in the set is not the cardinal number of the set. The basic concepts of the definition and determination of count and measure numbers form the cornerstone of the foundation of applied mathematics, probability and statistics, science and engineering, and economics. Ex. Application of the measuring method. Given a square box of chocolates, we want to know how long or the length of a side of the box. Using the measuring method, we place a one-foot long ruler marked with oneinch increments against the box such that the zero end of the ruler is at the left end of a side of the box. We now locate the point on the ruler that matches the end point on the right end of the side of the box. Suppose the number at the matching point on the ruler is 9.5 without units. (A point has no length.) The number 9.5 is the right end-point of the interval [0, 9.5] which has a length of 9.5 inches on the ruler. Then we can state that the length of a side of the box is 9.5 inches. Ex. Time measurement of a racer Starting time is established by pressing stopwatch; starting time is zero. Finishing time is established by pressing stopwatch; finishing time of 1.23 min. (say) is an ordinal number. Race time (length of time of race) of 1.23 min. is a (cardinal) measure number.

In general, any measuring instrument, e.g., a speedometer, gives a number that is the endpoint of an interval, and it is its distance from zero on the number line (length of the interval) that is actually the measure of quantity. A common mistake is to take the endpoint itself as the measure of quantity, i.e., a cardinal number. The end-point itself is just a number on the number line or can be thought of as an “ordinal” number. Measure numbers are used to compare continuous quantities. 1.3 is less than 2.4, denoted as 1.3 < 2.4, because (or means) the distance on the number line that 1.3 is from 0 is less than the distance on the number line that 2.4 is from 0. In terms of the related intervals we have the relation [0, 1.3] < [0, 2.4]. The reason is not simply because 1.3 is to the left of 2.4 on the number line which is an ordinal relation. Usually in mathematics books the relation 1.3 < 2.4 is given or assumed or defined and not explained why it is so. In pure mathematics the relation 1.3 < 2.4 has no physical meaning. Important examples of measure numbers Probability A probability is a measure number that represents how much certainty of occurrence of an event. The set of all probability values is equal to the set of all real numbers between 0 and 1 inclusive. Probability values can be expressed as decimals, fractions, and percents. A probability scale is a line segment or interval [0, 1], or equivalently 0% to 100%. The probability of an event is the number used to describe “how certain” we are that the event will occur. A probability p is an example of a measure number, and therefore, a cardinal number. A probability is a measure of how much certainty (how certain or how likely). A probability gets its measure of how much certainty by its length or distance from zero. The larger the length or distance p is from zero, the larger p is, the greater the certainty. A probability as a measure number is related to the interval [0, p] as follows: M([0, p]) = p An experiment is a process whereby an observation is made. A sample space is a set whose elements represent all possible outcomes of an experiment. Sample space S = {o1, o2, …, oN} where oi denotes outcome or sample point. An event is a subset of the sample space. Event A = {α1, α2, …, αn} where αi denotes outcome or sample point The probability of any event A, a subset of sample space S, is equal to the sum of all the “weights” assigned to the sample points in A. This sum is called the measure of A or the probability of A and is denoted by P(A). The probability of the empty set {} is zero and the probability of S is 1.Therefore, 0 ≤ P(A) ≤ 1, P({}) = 0, and P(S) = 1. P(A) = P({α1, α2, …, αn}) = p where p is a real number and 0 ≤ p ≤ 1 The probability function is a real-valued set function.

p = 0.5 implies 50% certain that event A will occur. p = 0 implies A is an impossible event p = 1 implies A is a certain or sure event Probability values are used to compare certainties: p1 = 0.25 is less than p2 = 0.50, denoted as p1 = 0.25 < p2 = 0.50, because (or means) the length or distance on the number line that 0.25 is from 0 is less than the length or distance on the number line that 0.50 is from 0. In terms of the related intervals we have the relation [0, 0.25] ⊂ [0, 0.50]. The reason is not simply because 0.25 is to the left of 0.50 on the number line. Usually in mathematics books the relation p1 < p2 is given or assumed. Even though probabilities are usually calculated in terms of decimals, it might be better to finally report them as percents for the following reasons: 1) A probability expressed in decimal form of 0.4 is read “point four” which has a connotation of a “point” on the number line rather than a distance from zero. 2) A probability expressed in percent form of 40% is read “forty percent” which has a connotation of a length more than a point. 3) Just having different numerical forms for the end-point and length is good, so that 0.4 is the calculated end-point of the interval [0, 0.4], and its length is 40%. 4) People are more familiar with percents than decimals as expressions of parts of the whole. Ex. Four kinds of numbers: identification, ordinal, and cardinal (count and measure). Experiment: A coin is tossed twice (2 times) (count number) (ordinal numbers) There is a 1st toss of coin, followed by a 2nd toss of coin. Let 1 denote head on a toss and 0 denote tail on a toss. (identification nos.) Sample space: S = {(1, 1), (1, 0), (0, 1), (0, 0)} (identification nos.) st nd where ordered pair is (outcome of 1 toss, outcome of 2 toss). (ordinal numbers} Number of sample points in S: C(S) = 4 (count number) Let event A: Exactly 1 head. (count number) Number of heads associated with the sample points: (identification nos. (1, 1) – 2, (1, 0) – 1, (0, 1) – 1, (0, 0) – 0 & count numbers) Event A = {(1, 0), (0, 1)} (identification nos.) Number of sample points in A: C(A) = 2 (count number) Probability of A: P(A) = 2/4 = ½ measure of certainty (measure number) Assumption: The 4 sample points in S are equally likely. (count number) Chiropractic A pain index p is a number that represents (or relates to) a measure of pain. A pain scale is a line segment or interval [0, 10] that is 10 centimeters in length. A pain index is an example of a measure number, and therefore, a cardinal number. M([0, p]) = p, pain index as a measure number is related to the interval [0, p]. The patient makes a slash through the pain line segment as to the level of pain felt. The chiropractor uses a metric ruler to determine the number on the ruler at the slash. The distance the ruler number is from zero will be the pain number that measures pain.

Let set A be a person who, of course, has a nervous system (set of nerves). Pain function of person A is equal to the pain index p. Notationally P(A) = p where p is a real number and 0 ≤ p ≤ 10 A pain function is a real-valued set function. Pain intensity: no pain, very mild, moderate, fairly severe, very severe, and the worst imaginable. Several corresponding pain index values are the following: p = 4.2 implies moderate pain p = 0 implies no pain p = 10 implies worst possible pain Suppose the pain index scale is restricted to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The patient now writes down one of these eleven numbers for the level of pain felt. This scale is a subset of the real numbers R between 0 and 1 inclusive, i.e., [0, 10]. Everything about measure numbers applies to any number in any subset of R. A measure number represents the distance (or length) that a real number is from zero. Remember that measure numbers are concerned with “how much.” There are many other examples where the numerical scale is restricted to a subset of R, say the rational numbers P, for purposes of simplification. Ex. Clothes sizes: shoes (8 ½), coat, shirt, dress, hat Ex. Attraction ratings: movies (3 ½ stars), TV shows, beauty (she’s a 10) Coefficient of determination The coefficient of determination, denoted ρ2, is a measure of the strength of the linear relationship (association) between two random variables X and Y, where ρ2 is a real number and 0 ≤ ρ2 ≤ 1. The coefficient of determination is an example of a measure number, and therefore, a cardinal number. M([0, ρ2]) = ρ2, coefficient of determination as a measure number is related to the interval [0, ρ2]. 100x ρ2 % of the variation in the values of Y is accounted for by a linear relationship with X. ρ2 = 0.5 implies 50% of the variation in the values of Y is accounted for by a linear relationship with X. ρ2 = 0 implies no linear relationship (no correlation) ρ2 = 1 implies perfect linear relationship (perfect correlation) Once again the distance from zero is the actual measure of strength of the association. Vector analysis A vector quantity is any physical quantity that has both magnitude a and direction. A vector is represented geometrically by a directed line segment (an arrow) whose length corresponds to its magnitude and points in an appropriate direction. Ex. Force, velocity or acceleration of an object.

The magnitude (size) or length of a vector is an example of a measure number, and therefore, a cardinal number. M([0, a]) = a, magnitude as a measure number is related to the interval [0, a]. Summary A. Certainty, pain, linear relationship, and magnitude are important examples of continuous quantities that have associated intervals whose lengths are represented by measure numbers. B. The corresponding methods of probability, pain index, coefficient of determination, and vector analysis determine an end-point number of an interval that starts at zero. C. The length of the interval is represented by a measure number whose value or number is equal to the end-point number. D. In practice we seldom differentiate between the end-point number and the measure number; just as with discrete quantities and counting we seldom differentiate between the last ordinal number in the set and the count number of the set which have the same value. E. The differentiation between the last number in a set and the number representing the quantity of the set is necessary in order to understand the meaning of cardinal numbers. F. Recall that mathematics is almost exclusively about cardinal numbers. OPERATIONS OF NUMBERS Union and separation are the most important basic operations of things in the world. Union and separation are the basic principle for administrating the affairs of the world. The operation of separation undoes or is the inverse or opposite of the operation of union. Whenever any thing (e.g., a ship) is made, there is a union of parts, and when the thing is junk, there is a separation of parts. Whenever any event (e.g., a game) is held, there is a union of participants, and when the event is over, there is a separation of participants. The mathematical operations of addition and subtraction along with multiplication and division are quantitative representations or descriptions of union and separation of things. Addition and multiplication are each a union of cardinal numbers (quantities). Subtraction and division are each a separation of a cardinal number (a quantity). The numbers considered in this section are cardinal numbers. Four operations of mathematics: addition, subtraction, multiplication, and division. Addition is the primary operation; the other operations follow from addition. Addition and subtraction are inverse operations; subtraction undoes addition. Multiplication is repeated addition. Ex. 3 * 4 = 4 + 4 + 4 = 12 Multiplication and division are inverse operations; division undoes multiplication. Axioms and theorems An axiom is a statement that is assumed to be true and upon which theorems are proved by using logical deduction.

Axioms are properties or laws that are assumed or defined. Theorems are properties or laws that are proved or derived. A mathematical property or equivalently a law can refer to an axiom or a theorem. (Properties or laws can refer to either axioms or theorems.) Axioms should be statements that cannot be demonstrated in terms of simpler concepts. The standard approach is to present mathematical operations as definitions and their properties as axioms. Instead of mathematical operations being definitions and their properties being axioms, they could be theorems. Mathematical operations can be derived from appropriate corresponding set operations which are definitions. Properties of mathematical operations can be derived from appropriate corresponding properties of set operations. Properties or laws of set operations, on the other hand, are themselves theorems whose proofs are based on truth or membership tables. There would be a greater understanding of the applied nature of the mathematical operations and their properties if they were theorems. Addition The standard approach in mathematics is to just define that 1 + 1 = 2. Addition will be analyzed and explained in terms of the basic operation of union. Union means to combine, join, or bring together things. Uniting is a process of combining, joining, or bringing together things. Three types of union: physical, set, and numerical. Three types of union for discrete quantities: Physical union of discrete objects. Set union of disjoint sets. Numerical union or addition of count numbers. Physical union of discrete objects. Ex. A horse is put into a corral that already contains a horse, getting two horses. Set and numerical union of discrete quantities Set theory approach to adding count numbers: Given two disjoint sets A = {α} and B = {β} with corresponding count numbers C(A) = C({α}) = 1 and C(B) = C({β}) = 1. A ∪ B = {α}∪{β} = {α,β} set union of disjoint sets C({α}) + C({ β}) = C({α, β}) where the plus sign (+) is the union operator symbol used to unite or add count numbers. 1 + 1 = 2 numerical addition of count numbers Assumption: α and β do not “interact;” they must maintain their individual identities even though they are united. Individual things are united such that they become related in some sense, but they retain their individual identities. There cannot be any merging because we need to be able to count individual things.

In the physical world, when you unite an object with another object, you don’t always get two objects. Ex. Add one lamb to the cage of one lion, and you get one lion Ex. Add one male rabbit to the cage of one female rabbit, and you get more than two rabbits. Ex. Add one suicide bomber to the room of one political leader, and you get zero people. Duality assumption Two sets are united into a single set of individual members which are not merged. {α}∪{β} = {α,β} and 1 + 1 = 2 Unity assumption Two sets are united into a single set of individual members which are merged into a single member. {α}∪{β} = {α} and 1 + 1 = 1 Once again, the standard approach in mathematics is to state that 1 + 1 = 2 by definition. Ordinary mathematics is simply considering the most common situation or case. Most of the time in the physical world when we unite something to something else we get two things. Three types of union for continuous quantities: Physical union of continuous objects. Set union of disjoint line intervals. Numerical union or addition of measure numbers. Physical union of continuous objects. Ex. A quart of water is put into a container that already contains a quart of water, getting two quarts of water. Set and numerical union of continuous quantities Number line approach to adding measure numbers: Given two intervals or line segments A = [0. 1] and B = [0, 1] with corresponding lengths or measure numbers M(A) = M([0, 1]) = 1 and M(B) = M([0, 1]) = 1. Let C = (1, 2] ≡ {r | r is a real number and 1 < r ≤ 2} M(C) = M((1, 2]) = M([0, 1]) = M(B) = 1 Intervals B and C have equal lengths of 1. A ∪ C = [0, 1] ∪ (1, 2] = [0, 2] set union of disjoint sets Note that A ∪ B = [0, 1] ∪ [0, 1] = [0, 1] = A = B not set union of disjoint sets M([0, 1]) + M((1, 2]) = M([0, 2]) where the plus sign (+) is the union operator symbol used to unite or add measure numbers. 1 + 1 = 2 numerical addition of measure numbers Subtraction

Three types of separation for discrete quantities: Physical separation of discrete objects. Set separation of mathematical sets. Numerical separation or subtraction of count numbers. Physical separation of discrete objects. Ex. A horse is taken from a corral that contains two horses, leaving one horse. Set and numerical separation of discrete quantities Set theory approach to subtracting count numbers: Given set A = {α, β} and subset B = {β} with corresponding count numbers C(A) = C({α, β}) = 2 and C(B) = C({ β}) = 1. A ~ B = {α, β} ~ {β} = {α} set separation C({α, β}) – C({β}) = C({α}) where the minus sign (–) is the union operator symbol used to separate or subtract count numbers. 2 – 1 = 1 numerical subtraction of count numbers PROPERTIES OF OPERATIONS The standard approach is to present the properties of mathematical operations as axioms. Axioms The axioms listed below form a foundation for the study of algebra, where a, b, and c are real number constants. Commutative axioms: a + b = b + a and ab = ba Associative axioms: a + (b + c) = (a + b) + c and a(bc) = (ab)c Distributive axiom: a(b + c) = ab + ac Identity axioms: a + 0 = a and a(1) = a Inverse axioms: a + (-a) = 0 and a(1/a) = 1, where a ≠ 0 Equality axioms: If a = b, then a + c = b + c and ac = bc The properties of mathematical operations could be theorems instead of axioms by proving them using the theorems of the algebra of sets. Set approach to the commutative axiom of addition: a + b = b + a Given disjoint sets A and B with corresponding count numbers C(A) = a and C(B) = b. Commutative law for union of sets: A ∪ B = B ∪ A C(A ∪ B) = C(B ∪ A) C(A) + C(B) = C(B) + C(A) because A and B are disjoint sets a+b=b+a

PART II: VARIABLES AND THEIR RELATIONS CONSTANTS There are actually two types of “constants”: universal constants and constants. A universal constant is a number or a letter that always represents a special number. Ex. 7; 2/3; 0.15; π ≡ 3.1416; e ≡ 2.7183 A constant is a letter that represents only one number or a fixed quantity throughout a particular discussion, relation, or problem but changes from problem to problem. Constants are usually denoted by a, b, and c. Ex. a = 2; b = ¾; c = 4.13; n = 15; p = 0.78 Constants are needed to state the axioms of mathematical operations. Ex. Distributive axiom: a(b + c) = ab + ac where a, b, and c are real number constants. VARIABLES A variable is a letter that represents any number from a set of numbers. The set of all possible values of a variable is called the domain or range. Variables are often denoted by x, y, and z. Ex. x = 7; y = 3, 9, 1, or 5; z = a real no. between 0 and 1; i = an integer; r = a real no. The concept of a variable could be considered to be an extension of the concept of a constant and includes constants, which is necessary, enabling a variable to take on only one value. There are many mathematical problems that have a single solution. A variable is a constant if its domain contains only one number. A variable is not a constant if its domain contains two or more numbers; then a variable can vary and take on various values, being a changing quantity. Variables and constants represent numbers, while numbers represent quantities. Whereas variables may assume different values in one problem, constants assume different values only in different problems. It should be clearly understood that whether a quantity is a constant or a variable may (and very often does) depend on the particular discussion at hand. Two types of variables: discrete and continuous A discrete variable is a variable that can take on a count number for a discrete quantity. A continuous variable is a variable that can take on a measure number for a continuous quantity. Domain of a discrete variable is a subset of the integers Z; domain of a continuous variable is a subset of the real numbers R. Generally in mathematics variables denote by letters and describe by words quantities of interest that are studied in word and real-world problems. Examples: Discrete variable n: number of children; n is a whole number, e.g., n = 3.

Discrete variable n: number of pages; n is a natural number, e.g., n = 238. Continuous variable h: height of person; h is a positive real number, e.g., h = 68 inches. Continuous variable w: weight of person; w is a positive real number, e.g., w = 140 pounds. RELATIONS Algebra and higher mathematics are all about variables and their relations. Relations between variables consist of equations, formulas, and functions. An equation is a mathematical statement that two quantities are equal using an equal sign. The solution set of an equation is the set of all values for the variable which cause the equation to be a true statement. An equation can have one or more solutions. Ex. A linear equation in variable x has the form ax + b = 0 where a and b are real number constants and a ≠ 0. The solution set is {-b/a}. The variable x is a constant. Five steps in solving word or real-world problems: 1. Declare the variables using appropriate letters. 2. Determine the equations using given relationships. 3. Solve the equations using mathematical operations. 4. Answer the question using the solution. 5. Check the work using careful review. A formula is an equation which relates two or more variables. Ex. The formula d = rt describes the relationship between distance, rate, and time. A function from set A to set B is a correspondence (or rule) that assigns to each value of the variable x in set A a unique value of the variable y in set B. A function describes a relationship between two variables: y is a function of x. Variable x is called the independent variable; variable y is called the dependent variable. A function is a relation between an independent variable x and a dependent variable y such that for each distinct value of x there is exactly one value of y. Set A of all values of x is called the domain; set B of all values of y is called the range. A linear function is any function of the form y = ax + b where a and b are real number constants and a ≠ 0. If a = 0, then y = b is a constant function, not a linear function. The graph of a linear function is a straight line with slope a and y-intercept b. Ex. The linear function relating time t (hours) and distance d (miles) with rate r = 60 mph is d = 60t where t and d are positive real numbers. The graph of this function is a straight line with slope 60 and y-intercept 0. For t = 3 hours, d = 60(3) = 180 miles. PROBABILITY Random variables The most important concept in probability and statistics is a random variable. Random variables are usually denoted by X, Y, and Z

Recall that sets are usually denoted by A, B, and C; constants are usually denoted by a, b, and c; variables are usually denoted by x, y, and z. A random variable X is a variable that has a probability distribution (function). The set of all possible values of a random variable is called the domain. A random variable is a constant if its domain contains only one number; This degenerate case rarely occurs in practice, not like a variable. Two types of random variables and probability distributions: discrete and continuous. Discrete distributions: binomial, negative binomial, geometric, hypergeometric, Poisson. The domain of a discrete random variable is a subset of the integers Z. Continuous distributions: uniform, triangular, normal, lognormal, exponential, gamma, chi-square, Weibull, Pareto. The domain of a continuous random variable is a subset of the real numbers R. Philosophy Proposition: The physical universe or nature is deterministic. Determinism: Every event has a cause; law of cause and effect (or law of karma). Einstein: “God does not play dice with the universe.” Chaos Chaos is apparent randomness from extremely complex behavior occurring in a deterministic process due to excessive sensitivity of the result (or event) to small changes in initial conditions. Ex. Flipping a coin, which is a deterministic process or experiment. There are theoretical and practical limitations to our knowledge of nature. Our uncertainty is due to our limitations to our knowledge of nature. Definition of probability A probability is a measure number (between 0 and 1 inclusive) that represents or describes how much certainty a person has of the occurrence of a particular event (or observation), which is assigned based on the theoretical or empirical information held at some time and the assumptions that are made. (A probability value of 0 means we believe that the occurrence of the event is impossible, whereas 1 means certain.) A probability can be viewed as a “weight” (another measure number). Determination of probability There are two objective approaches to the determination of an assignment of probability to an event: classical approach and relative frequency approach. A third approach is the subjective approach. The order of presentation of the three approaches is their order of historic development and also their order of strategic consideration. That is, the classical approach was developed first and also might be considered first in the determination of an approach for making an assignment of probability; the two other approaches follow in order.

Classical approach by B. Pascal in 1654 (theoretical information): If an experiment can result in any one of N different equally likely outcomes, and if m of these outcomes correspond to event A, then the probability of event A is P(A) = m / N Application: Games of chance where assumption of equally likely is often made. Ex. Experiment is throwing a die; event A is even number of dots; assumption is fair die. P(A) = 3/6 = ½ = 0.5 Relative frequency approach by J. Venn in 1866 (empirical information): If an experiment can be repeated a large number of times n under similar conditions, and if f denotes the frequency of occurrence of event A, then the probability of event A is P(A) = f / n Application: Engineering and biology where repetition of experiment is often possible. Ex. Experiment is testing a tire; event A is defective tire; information is 100 tires are tested, and there are 10 defective tires; assumption is similar tires. P(A) = 10/100 = 1/10 = 0.1 Subjective approach by R.P. Ramsey in 1926 (theoretical or empirical information): Indirect information includes physical features of the experiment, scientific reasoning, our laws of nature, geometrical considerations, and analogy. The probability of event A is P(A) = Subjective percentage Application: Geology and economics where repetition of experiment is often not possible. Ex. Experiment is drilling a well; event A is discover oil; information is geological study shows favorable conditions for oil; assumption is favorable geologic analog. P(A) = 75% = 0.75 “Which assignment of probabilities to the simple events should be made is not a mathematical question, but one that depends upon our assessment of the real-world situation to which the theory is to be applied” (Goldberg, 1960). Pure mathematics is also not concerned with the physical meaning of probability. Concept of probability used in statistics

Most statistics books embrace the limiting relative frequency concept of probability as the definition of probability. Limiting relative frequency concept by R. von Mises in 1919: Consider a sequence of repetitions of the same experiment under identical conditions. Let fn denote the number of occurrences of the event A in the first n repetitions of the experiment. The ratio fn/n then gives the relative frequency of occurrence of event A in the first n repetitions. The probability of an event A is P(A) = limn→∞ (fn / n) that is, the limiting relative frequency of occurrence of event A in the long run. The limiting relative frequency concept of probability is an abstract concept and not a working concept that can actually be applied. The same experiment cannot be repeated under identical conditions indefinitely. The concept of repeating the same experiment under identical conditions forever violates two fundamental properties of nature: change and noneternal; everything in nature is continuously changing and eventually ending. Regarding this concept, John Keynes said: “In the long run we shall all be dead.” Under determinism identical conditions would produce identical events. It would take different conditions to produce different events. In nature conditions are always changing, and, according to chaos theory, this is what produces the apparent randomness of events. There is also the question of convergence to some constant limiting value. Ross (1988) states that the limiting relative frequency concept should not be the definition of probability but rather a theorem in probability theory: “How do we know that fn/n will converge to some constant limiting value that will be the same for each possible sequence of repetitions of the experiment?... Proponents of the [limiting] relative frequency definition of probability usually answer this objection by stating that the convergence of fn/n to a constant limiting value is an assumption, or an axiom, of the system. However, to assume that fn/n will necessarily converge to some constant value seems to be a very complex assumption. For, although we might indeed hope that such a constant limiting frequency exists, it does not at all seem to be a priori evident that this need be the case. In fact, would it not be more reasonable to assume a set of simpler and more self-evident axioms about probability and then attempt to prove that such a constant limiting frequency does in some sense exist? This latter approach is the modern axiomatic approach to probability theory that we shall adopt in this text (Ross, 1988).” Axioms of probability by A.N. Kolmogorov in 1933 Consider an experiment whose sample space is S. For each event A of the sample space S we assume that a measure number P(A) is defined and satisfies the following three axioms.

Axiom 1 If A is any event, then 0 ≤ P(A) ≤ 1 The probability of A is a real number between 0 and 1 inclusive. Axiom 2 If S is the sample space – the set of all possible outcomes of an experiment, then P(S) = 1 The event S is a sure or certain event. Axiom 3 If A1, A2, A3,… is a sequence of mutually exclusive events, then P(A1 ∪ A2 ∪ A3 ∪ …) = P(A1) + P(A2) + P(A3) + … The probability of at least one of these events occurring is just the sum of their respective probabilities. It is important to realize that Axiom 3 is in terms of the basic concept of the union of mutually exclusive sets and addition of measure numbers. The theory of probability is based on and developed from the axioms of probability. Conditional probability The conditional probability of event B given that event A has occurred with P(A) > 0 is defined by P(B | A) = P(A ∩ B)/P(A) Given that A has occurred, event A becomes the new reduced sample space. Independence Two events A and B are independent if P(B | A) = P(B) and are dependent otherwise. That is, event B is independent of event A if knowledge that A has occurred does not change the probability that B occurs. If two events are mutually exclusive, then they are dependent. Rules of probability Addition rule If A and B are any two events, then P(A ∪ B) = P(A) + P(B) – P(A ∩ B) Special addition rule If A and B are mutually exclusive events, then

P(A ∪ B) = P(A) + P(B) Complement rule If event A’ is the complement of event A, then P(A’) = 1 – P(A) Multiplication rule If A and B are any two events, then P(A ∩ B) = P(A)P(B | A) Special multiplication rule If A and B are independent events, then P(A ∩ B) = P(A)P(B) Another special multiplication rule If A and B are mutually exclusive events, then P(A ∩ B) = 0 Methods of probability There are two general probabilistic methods for solving probability problems in probabilistic analysis: analytic probabilistic method and Monte Carlo simulation method. Analytic probabilistic method A probabilistic method that uses mathematical equations from probability theory. Analytic methods: Stochastic processes, Markov chains, queuing theory, game theory, decision theory, risk analysis, dynamic programming, reliability theory, time series analysis, and actuarial analysis. Monte Carlo simulation method A probabilistic method that uses repeated sampling from computer application. The simulation method is an alternative to the analytic method when the probability problem is mathematically untractable. STATISTICS “The science of statistics is essentially a branch of Applied Mathematics, and may be regarded as mathematics applied to observational data” (R.A. Fisher, 1967). Statistics is the science that deals with the collection, classification, analysis, and interpretation of a set of numerical data.

The collected data set is usually a set of cardinal numbers. Two types of numerical data: Discrete or count data and continuous or measured data. Ex. Measurements of porosity and permeability of reservoir rocks. The science of statistics is used to describe, relate, and compare populations. A population is the set of all observations (numbers) with which we are concerned. Descriptive statistics Descriptive statistics is the study of techniques for describing a given population of data. The collected data set is the population of interest. Each observation or number in a population is a value of a variable x. A population is characterized by a variable x with a finite distribution. Statistical techniques: Bar and pie charts, graphs, tables, and population parameters. A population parameter is a parameter that describes a population distribution. A parameter is a constant or fixed value. Ex. Population mean, median, and mode which are measures of central tendency; population range, variance, and standard deviation which are measures of variation. Population mean and population standard deviation are denoted by μ and σ. Population mean, variable mean, and distribution mean are equivalent. Inferential statistics Inferential statistics is the study of procedures for making inferences about a population on the basis of a sample by utilizing the field of probability. A sample is a subset of a population. The collected data set is a sample from the population of interest. Each observation or number in a population is a value of a random variable X. A population is characterized by a random variable X with a probability distribution. A population distribution is modeled by a probability distribution. Inferences Inferences are made about the population parameters or distribution. Two types of inference: Statistical estimation and tests of hypotheses. Statistical estimation: Inference is an estimate of a population parameter. Two types of estimation: Point (single value) and interval (continuous set of values). Ex. Point estimate or confidence interval estimate of a population mean. Tests of hypotheses: Inference is a test of hypothesis of a population aspect. Ex. Test of the hypothesis that a population has a lognormal distribution. Lognormal distribution: A population (or random variable) has a lognormal distribution if the natural logarithm of the population has a normal distribution. Sample statistics A random sample is a set of n independent and identically distributed random variables X1, X2, …, Xn each having the same probability distribution as the population X, where n is called the random sample size.

A statistic is a function of and calculated from a random sample. Ex. Sample mean, median, mode, range, variance, and standard deviation. ¯ and S. The sample mean and sample standard deviation are denoted by X A statistic is a random variable and has a probability distribution. Population parameters are constants; sample statistics are random variables. The probability distribution of a statistic is called a sampling distribution. A statistic or sampling distribution has a mean and standard deviation, which are parameters. An estimator is a statistic that is used to estimate a population parameter. A test statistic is a statistic that is used to test a hypothesis of a population aspect. Sampling distribution of the sample mean The sample mean X ¯ = ΣXi/n is a random variable. The sample mean has a mean and standard deviation, which are parameters. Thm. The mean of the sample mean is equal to the population mean. Thm. The standard deviation of the sample mean is equal to the population standard deviation divided by the square root of n. The Central Limit Theorem gives the sampling distribution of the sample mean: Given a population with any distribution, the sampling distribution of the sample mean is approximately a normal distribution if n is large, at least 30. This amazing result is the most important theorem in statistics. Thm. Given a population with a normal distribution, the sampling distribution of the sample mean is exactly a normal distribution for all n. Any normal distribution can be transformed into the standard normal distribution. The standard normal distribution is the normal distribution with mean 0 and standard deviation 1. The standard normal distribution is tabled to give z percentiles. Normal random variable and probability distribution are most important in statistics. ¯ is an estimator of the population mean μ. The sample mean X Statistical methods Estimation theory, tests of hypotheses, regression and correlation, analysis of variance, design of experiments, sampling techniques, sample surveys, nonparametric statistics, multivariate analysis, factor analysis, discriminate analysis, Bayesian statistics, quality control, and geostatistics (kriging). Parametric statistics Statistical methods that are based on the assumption of a normal distribution. Nonparametric (or distribution-free) statistics Statistical methods that are based on no assumption of the population distribution. Nonparametric methods: Nonparametric tests and rank correlation coefficient. Nonparametric methods are an alternative to parametric methods when the normality assumption cannot be made. Application Every discipline has its favorite statistical methods that are most often used.

Ex. Geology favors regression and correlation, multivariate analysis, and kriging. REFERENCES Fisher, R.A., 1967, Statistical methods for research workers: New York, N.Y., Hafner, 356 p. Goldberg, S., 1960, Probability, an introduction: Englewood Cliffs, N.J., Prentice-Hall, 322 p. Newman, J.R., 1956, The world of mathematics, 4 vol.: New York, N.Y., Simon and Schuster, 2537 p. Ouspensky, P.D., 1970, Tertium organum: New York, N.Y., Vintage Books, 311 p. Ross, S., 1988, A first course in probability, 3rd ed.: New York, N.Y., Macmillan, 422 p.