課程名稱︰機率與統計
課程性質︰系訂必修
課程教師︰鍾嘉德 張時中 葉丙成 林茂昭
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰97.6.19
考試時限（分鐘）：110
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
1. Describe the Central Limit Theorem. (5%)
2. Suppose that X and Y are two independent zero mean Gaussian random varibles
with variaces (σX)^2 and (σY)^2.
(a) What is the joint probability density function of (X,Y)? (5%)
(b) What is the probability density function of X-Y? (5%)
3. Suppose that X and Y are two zero mean Gaussian random varibles with
covariances Cov[X,X]=(σX)^2 , Cov[X,Y]=σXY , and Cov[Y,Y]=(σY)^2.
What is the probability density function of X+Y? (10%)
4. Consider two random variables X and Y.
(a) Describe the condition that X and Y are uncorrelated implies that X and
Y are independent. (5%)
(b) Suppose Cov[X,Y]=0. Is it necessary that X and Y are independent? Please
provide your reason. (5%)
5. Describe the condition that MAP test is equivalent to the maximum likeli-
hood test. (5%)
6. Suppose that we know E[X]=μX in advance. Is
n
(1/n) Σ (Xi-μX)^2
i=1
an unbiased estimate of Var[X]? Prove your answer. (5%)
7. X and Y are two random variables with a joint probability density function
as
┌cxy , 1≦x≦y≦2;
f (x,y)=│
XY └0 , otherwise.
(a) c=? (5%)
(b) Are X and Y independent? Explain why. (5%)
(c) Derive the conditional probability distribution function F (y|x). (5%)
Y|X
(d) Var[E[Y|X]]=? (5%)
(e) Correlation coefficient ρ(X,Y)=? (5%)
(f) Let U=X+Y and W=3X+4Y. Find the covariance matrix of [U W]'. (5%)
8. You are counting the numbers of buses, cars and bicycles passing by where
you stand. Let KB, KC and Kb be the numbers you get during time [0,T],
which are independent Poisson random varibles with parametersλB, λC and
λb.
(a) Prove that K=KB+KC+Kb is also a Poisson with parameterλ=λB+λC+λb.
(Hint: Use moment generating function) (10%)
(b) Let event A be the event that "when a vehicle passes by, it is a bicy-
cle". P[A]=? (5%)
9. Let X be binomial random varible with parameters (n,p) and n=3. For the
following hypothesis test
Ho:p=1/2 vs Ha:p=2/3
we reject Ho as X=0 or 3. Please calculate
α＝the probability of making a type I error
β＝the probability of making a type II error. (10%)
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