Description
1. Find the moment generating function of the following random variables
(i) Binomial(n, p);
(ii) Poisson(λ);
(iii) Gamma(α, β), i.e., the pdf is given by
f(x) = β
α
Γ(α)
x
α−1
e
−βx, x > 0
2. (a) In the casino game roulette, the probability of winning with a bet on red is p = 18/38.
Let X be the number of winning bets out of 100 independent bets that are placed.
Find P(X > 50) approximately.
(b) Let {Xi
, 1 ≤ i ≤ 16} be a random sample form a distribution with pdf
f(x) = 3x
2
, 0 < x < 1. Approximate P(X <¯ 0.5).
3. Let X1 and X2 be a random sample from N(µ, 1).
(i) Find P(X1 − X2 < 1);
(ii) Prove that X1 − X2 and X1 + X2 are independent.
4. If X and Y are independent standard normal random variables. Show that X/Y has a
t-distribution with 1 degree of freedom, which is also called the Cauchy distribution.
5. Let {Xi1, · · · , Xini
} be a random sample from N(µi
, σ2
), i = 1, 2. Assume that the random
samples are independent. Prove that
S
2
1
/S
2
2
has an F-distribution with n1 − 1 and n2 − 1 degrees of freedom, where S
2
i
, i = 1, 2 are the
sample variance of the random samples.
6. Let X1, X2, · · · , Xn be a random sample from the uniform distribution U(0, 1). Find the
pdf of the ith smallest order statistic X(i) and its expectation and variance.
7. Let X1, · · · , Xn be a random sample from N(µ, 1). Define
X¯
k =
1
k
X
k
i=1
(Xi − µ) and X˜
k =
1
n − k
Xn
i=k+1
(Xi − µ)
For 1 ≤ k ≤ n − 1,
(i) What is the distribution of X¯
k + X˜
k?
(ii) What is the distribution of kX¯ 2
k + (n − k)X˜ 2
k
?
(iii) What is the distribution of kX¯ 2
k
/((n − k)X˜ 2
k
)?