課程名稱︰機率導論
課程性質︰數學系大二必修
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/05/21
考試時限（分鐘）：70
試題 :
1. (25 points) The random variables X and Y are jointly continuously distributed
with joint density given by
-3 -y -2 -2y
╭ x e + x e x ≧ 1, y ≧ 0
f(x, y) = ╯
╰ 0 else.
a. (6 points) Compute the joint distribution function F(x, y) = P(X ≦ x, Y
≦ y).
b. (6 points) Compute the density f (y) of Y alone.
Y
c. (6 points) Compute the conditional probability density f (x|y).
X|Y
d. (7 points) Compute P(X ≦ 2|Y = 1).
2. (30 points) Consider n independent and identically distributed random
variables X , ..., X with distribution function F(x). Write
1 n
1 n
F (t) =─ Σ 1 , t∈|R
n n i=1 {X ≦t}
i
and
n
N (t) = Σ 1 .
n i=1 {X ≦t}
i
a. (10 points) Is N (1) a binomial random variable? If the answer is YES,
n
N (1) ~ Binomial(n, p) and determine p. Please give reason to justify your
n
answer.
b. (10 points) Show that F (1) converges in probability and find its limit.
n
＿
c. (10 points) Show that for any t∈|R, √n ( F (t) - m(t) ) converges to a
n
2 2
normal distribution N(0, σ (t)) for function m(t) and σ (t) which you
specify.
3. (24 points) Let the point (X, Y) be uniformly distributed over the half disk
2 2
x + y ≦ 1 where y ≧ 0.
2
a. (12 points) If you observe X, find θ(x) which minimizes E[(Y-θ(X)) ].
2
b. (12 points) If you observe Y, find η(y) which minimizes E[(X-η(Y)) ].
4. (25 points) Let X , X , ..., X be independent normal random variables with
1 2 n
2 n
mean μ and variance σ . Consider random variable Y where Y = Σ a X . Here
i i i=1 i i
the a are scalars. Use moment generating function to show that Y is normally
i
distributed, and find its mean and variance in terms of moment generating
function.
2 2
2 υt +τ t /2
When X ~ N(υ, τ ), its moment generating function is m (t) = e .
X