課程名稱︰機率導論
課程性質︰大三必修
課程教師︰陳宏
開課學院：理學隊
開課系所︰數學系
考試時間︰Tuesday: 1:20-3:10pm, November 14th, 2006
是否需發放獎勵金：是
試題 :
Introductory Probability
Midterm
Tuesday 1:20-3:10pm, November 14th, 2006
1. (30 points) Suppose the probability a married man votes is 0.35, the
probability a married woman votes is 0.40, and the probability a woman
votes given that her husband does is 0.6. Find the probability that
(a) (10 points) both a husband and wife vote,
(b) (10 points) a man votes given that his wife does not,
(c) (10 points) a woman votes given that her husband does not.
2. (40 points) If n poeple, among whom are Al and Betty, are standing in a row
(a) (20 points) what is the probability that exactly r, rε{0,1,...,n-2},
persons are standing in between Al and Betty?
(b) (20 points) If they are standing in a ring instead of a row, show that
the probility is independent of r and is 1/(n-1) (here we are
considering the arc from Al to Betty in the counterclockwise direction.
3. (50 points) Let X be a continous random variable with density
f(x) = 0.5 exp(-|x|), xεR
(a) (15 points) Calculate P(X<1).
(b) (15 points) Calculate E(X).
(c) Let Y=X^2. Find the distribution function of Y.
4. (50 points) Let X have a binomial distribution with n trials and success
probability p. Let t be an arbitrary real number. Find E[exp(tX)].
5. (50 points) Suppose the number of customers arriving in a store during a
typical hour is Poisson distributed with parameter λ. Let p be the typical
fraction of customers that are female. In other words, conditioning on the
event that exactly n customers come during a given hour, the probability
that k of them are female is
n! k n-k
───── p (1-p) .
k!(n-k)!
(a) (30 points) Show that the number of female customers arriving in a
given hour is Poisson distributed with parameter pλ.
(b) (20 points) Show that the event that no females arrive during the hour
is independent of the event that no male arrives during the hour.
6. (50 points) Determine a to minimize E|X-a| where X is a nonnegative conti-
nous random variable with density function f(x)=exp(-x) where x>=0.
7. (30 points) A population P has N member and that an ordered sample of size
n is chosen from it with replacement. Assume all outcomes are equally
likely. Suppose N and n are both allowed to vary. Show that the probability
of no repetitions in the sample approaches 1 if and only if n^2/N goes to 0
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