課程名稱︰機率導論
課程性質︰數學系必修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/6/15
考試時限（分鐘）：110分鐘
試題 :
Introduction to Probability Finial Examination 15 June 2017
1.(20 pts.) True or False. Give a brief reason or a counter example.
(a) If X and Y are Poisson random variables with respective parameters λ
1
and λ (λ ,λ ＞0), then X+Y is a Possion random variable with
2 1 2
parameter λ +λ .
1 2
(b) If X is a random variable with E[X]=50, then P{X＞100}≦0.5.
(c) Suppose that X and Y are both Bernoulli randon variables. Then X and Y
are independent if and only if Cov(X,Y) = 0.
(d) Suppose that X ,...,X have a multivariate normal distribution. Then X
1 n 1
,...,X are independent if and only if Cov(X , X ) = 0 when i≠j.
n i j
(e) Let X ,X ,... be a sequence of random variables and c is a constant.
1 2
If for each ε＞0, P{|X －c |＞ε} → 0 as n → ∞, then E[X ] → c as
n n
n → ∞.
2. Suppose that random vector (X,Y) has a joint probability density function
2 2 2 2
/ c √(1－x －y ) , if x ＋y ≦1, x≧0, y≧0,
(pdf) given by f(x,y) =
\ 0 , otherwise
for some constant c.
(a) (5 pts.) Find the value of the constant c and marginal pdf of Y.
2 2 -1 Y
(b) (5 pts.) Find the joint pdf of R = √(x ＋y ) and Θ = tan (———).
X
Compute E[R].
＊ 2
(c) (5 pts.) Find g (Y) that minimizes E[(X-g(Y)) ] over functions g on R.
3. Suppose that an experiment can result in one of r possible outcomes, the ith
r
outcome having probability p ,p ,...,p , Σ p = 1. If n independent
1 2 r i=1 i
experiments are performed, let N denote the number of times outcome i
i
occurs.
(a) (5 pts.) Find Cov( N , N ) .
i j
(b) (5 pts.) Find the mean and variance of the number of outcomes that do
not occur.
4. Suppose that X ,X ,...,X are independent uniform (0,1) random variables.
1 2 n
Let X ,X ,...,X be the order statistics.
(1) (2) (n)
(a) (5 pts.) Find the joint pdf of X and X , 1≦i＜j≦n.
(i) (j)
(b) (5 pts.) Let R = X －X denote the range and M = [X ＋X ]/2 the
(n) (1) (n) (1)
midrange of X ,X ,...,X . Find the joint pdf of R and M.
1 2 n
5. Suppose the distribution of Y, conditional on X = x, is Normal(x,x) and that
the marginal distribution of X is uniform(0,1).
(a) (5 pts.) Find E(Y), Var(Y), and Cov(X,Y).
(b) (5 pts.) Find the joint moment generating function of X and Y.
6. (a) (4 pts.) State the Central Limit Theorem for i.i.d. samples.
(b) (8 pts.) A six-sided die is continually rolled until the total sum of
all rolls exceeds 380. Using Central Limit Theorem to approximate the
probability that at least 100 rolls are necessary.
7. (8 pts.) State the Weak Law of Large Numbers for i.i.d. samples, and give a
proof when the variance exists.
2
8. Suppose X ,...,X are i.i.d. Normal(μ,σ ) random variables. Let
1 n
T n
X = ( X , X ,..., X ) be a random vector, b ∈ R be a constant vector
1 2 n
and A be an invertible n ×n matrix.
(a) (5 pts.) Identify the probability distribution of Y = AX + b.
(b) (5 pts.) Identify the probability distribution of sample variance
2 1 n _ 2 _ 1
S = ——— Σ (X － X ) , where X = —— ( X ＋...＋ X ) is the
n-1 i=1 i n n n 1 n
sample mean.
2 2
(c) (5 pts.) Find E[S ] and Var (S ).