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課程名稱︰隨機程序及應用
課程性質︰選修
課程教師︰葉丙成
開課學院：電資學院
開課系所︰電信所
考試日期（年月日）︰2009.10.21
考試時限（分鐘）：180 + 20(延長)
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.(2%) Please write down your name on the top of every answer sheet.
2.Suppose X is a normal random variable with zero mean and unit variance.
Please plot the CDF of Y where Y=g(X) and indicate some special points which
can be identified with c. The CDF of X, Φ(x), can be used.
(a) (3%) ┌x-c , x＞c
│
g(x) = ┤ 0 , -c≦x≦c
│
└x+c , x＜-c
(b) (3%) Please find the PDF of Y=g(X) as defined in (a).
3.Please prove the mean and variance of the following distributions is as
given. Detail derivation is needed to get full credit.
(a) (3%) Geometric distribution with parameter p has mean 1/p and variance
(1-p)/p^2.
(b) (3%) Exponential distribution with rate λ has mean 1/λ and variance
1/λ^2.
4.Please prove the moment generating functions of the following is as given.
Detail derivation is neded to get full credit.
(a) (4%) Exponential distribution with rate λ has MGF λ/(λ-t) with t＜λ.
(b) (4%) Normal distribution with mean μ and variance σ^2 has MGF
exp{tμ+ [(σt)^2]/2 }.
5.Please calculate the following integrations. Detail derivation is needed to
get full credit.
(a) (4%) ∞
∫ (2x^2+5x+1)e^(-x^2)dx
-∞
(b) (4%) ∞ (λ^α)(x^α+1)(e^-λx)
∫ ------------------------dx
0 α!
6.The joint pdf of X and Y is given by
[e^(-x/y)][e^y]
f(x,y)= --------------- , 0＜x＜∞ , 0＜y＜∞
y
(a) (3%) Please find E[X|Y=y].
(b) (3%) Please find Var(X|Y=y)
7.(5%) Let X and Y be indpendent random varibles with means μx and μy and
variances σx^2 and σy^2. Please express Var((X-1)Y) in terms of μx, μy,
σx^2 and σy^2.
8.Box one contains 1000 bulbs and there are 10% of the bulbs are defective.
Box two contains 2000 bulbs and there are 5% of the bulbs are defective.
Two bulbs are picked from a randomly selected box.
(a) (3%) Find the probability that both of the bulbs are defective.
(b) (5%) Assuming that the two bulbs are both defective, please find the
probability that they come from box one.
9.A box contains n white and m black marbles. Let X represent the number of
draws needed for the rth white marble.
(a) (2%) If sampling is done with replacement, show that X has a negative
binomial distribution with parameters r and p=n/(n+m).
(b) (4%) If sampling is done without replacement, then show that
m+n-k
( )
k-1 n-r
P(X=k) = ( ) ---------, k=r,r+1, ... ,m+n
r-1 m+n
( )
n
(c) (4%) For a given k and r, show that the probabilty distribution in (b)
tends to a negative binomial distribution as n+m→∞. Thus, for large
population size, sampling with or without replacement is the same.
10.(5%) Suppose that X and Y are independent, identically distributed, geome-
tric random variables with parameter p. Show that
1
P( X=i | X+Y=n) = ----- , i=1,2, ... ,n
n-1
11.Let A and B be events with P(A)>0 and P(B)>0. We say that an event B sug-
gests an event A if P(A|B)>P(A), and does not suggest event A if
P(A|B)<P(A).
(a) (3%) Show that B suggests A if and only if A suggests B.
c c
(b) (3%) Assume that P(B )>0. Show that B suggests A if and only if B
does not suggest A.
(c) (4%) We know that a treasure is located in one of the two places, with
probability β and 1-β, respectively, where 0<β<1. We search the first
place and if the treasure is there, we find it with probability p>0.
Show that the event of not finding the treasure in the first place
suggests that the treasure is in the second place.
12.Let X and Y be independent uniform random varibles from zero to one. Let
Z=X+Y and W=X-Y.
(a) (3%) Please find f(z).
(b) (3%) Please find f(w).
(c) (4%) Please show that Z and W are uncorrelated random varibles but not
independent random varibles.
13.There are two service counters in the post office. The service time at the
counter one and counter two are independent and exponentially distributed
with rate λ1 and λ2 respectively. When C comes to the post office, A and
B, have been at the counter one and two respectively. C waits behind them.
Please find
(a) (3%) The probability that A leaves last and the probability that B
leaves last.
(b) (3%) The probability that C leaves last.
(c) (5%) The expected time spent in the office the person who leaves last.
14.(5%) A waiter calls two taxis for a eight-people family. Assume the times
that the two taxis are exponentially distributed with rate λ1 and λ2.
Please calculate the expected time that the waiter needs to wait for both
the taxis to come.
PS.試題後有附上各種機率分配的p(x)和f(x)
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