推 t0444564 : 已收錄 05/24 01:20
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
--
2 2 1 ψxavier13540
給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system
╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則
╰x(0) = x° t
ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三
次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。
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