AS/2603/WKC/1 UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS Probability Theory Assignment 1 Due Date: 30 September 2006. 1. Let A, B, C are any three events, prove that P (A∪B∪C) = P (A)+P (B)+P (C)−P (A∩B)−P (A∩C)−P (B∩C)+P (A∩B∩C) 2. Consider a discrete random variable X which follows the probability distribution pi =
6 (πi)2
i = 1, 2, . . . .
Find E(X). 3. In a nuclear plant, accidents are mainly caused by Human error (H), Mechanical error (M) and Natural disaster (N). From past experience, in each year, the conditional probabilities that the nuclear plant will explode when there is human error, Mechanical error and Natural disaster are 0.01, 0.05 and 0.1 respectively. The occurrence probabilities of Human error, Mechanical error and Natural disaster are 0.02, 0.01 and 0.001 respectively and any two of the errors or disaster will not occur at the same time. Suppose unluckily that the nuclear plant has explosion what is the probability that it is due to human error? 4. Prove that the mean and variance of the Binomial distribution are given respectively by µ = np and σ 2 = npq. Here q = 1 − p. 5. Let X1 , X2 , . . . , Xn be n independent continuous random variables, each having a uniform distribution over (0, 1). Let Y = max{X1 , X2 , . . . , Xn }. Find the probability density function of Y . 6. A salesman has to meet a number of clients each day. ¿From past experience, the number of clients X follows the geometric distribution p(1 − p)i−1 ,
0 < p < 1,
i = 1, 2, . . . .
Moreover, for each client, the probability of getting a successful business agreement is q. Let Y be the number of successful business agreements he gets on a day. Find the probability distribution P (Y = y) and hence E(Y ).
1
7. (a) In a sample space, let A and B be the two events. Prove that if P(A)≥ 0.8 and P(B)≥ 0.5, then P(A ∩ B) ≥ 0.3. (b) Prove that for arbitrary events A1 , A2 , . . . , An of a sample space the following inequality holds: P (A1 ∩ A2 ∩ . . . An ) ≥ P (A1 ) + P (A2 ) + . . . + P (An ) − (n − 1) 8. Let a discrete random variable has a PDF of the form f (x) = c(8 − x) for x = 0, 1, 2, 3, 4, 5, and zero otherwise. (a) Find the constant c. (b) Find the CDF, F (x). (c) Find P (X > 2). 9. Find the moment generating function for the probability density function f (x) =
e−λ λx , x!
x = 0, 1, 2, . . . .
10. Let X and Y be two independent random variables taking values in [1, ∞). Let f (x) and g(y) be their probability density functions respectively. We assume that both f (x) and g(y) are positive and continuous functions and Z
∞
Y X
Z
f (x)dx =
E
g(y)dy = 1.
1
1
Define
∞
=
Z
∞ 1
Z
∞
Y X
1
y f (x)g(y)dxdy. x
Prove or disprove the following E where E(X) =
Z
E(Y ) E(X)
≥
∞
xf (x)dx and E(Y ) =
1
Z
∞
yg(y)dy.
1
11. Let f (x) be the probability density function of the lifetime X of a system and R(t) =
Z
∞
f (x)dx.
t
Suppose that lim x2 f (x) = 0 and E(X) < ∞,
x→∞
show that E(X) =
Z
0
2
∞
R(t)dt.
