课程名称︰机率导论
课程性质︰数学系大二必修
课程教师︰陈宏
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/04/23
考试时限（分钟）：120
试题 :
1. (40 points) An insurance policy reimburses a loss up to a benefit limit of
10. The policyholder's loss, Y, follows a distribution with density function
-2
╭ y , for y > 1
f(y) = ╯
╰ 0, otherwise
(a) (25 points) What is the expected value of the benefit, X, paid under the
insurance policy?
(b) (20 points) How do you write X in terms of a function of Y and 10?
2. (45 points) A company prices its hurricane insurance under the following
assumptions:
(i) In any calendar year, there can be at most one hurricane.
(ii) In any calendar year, the probability of a hurricane is 0.05.
(iii) The number of hurricanes in any calendar year is independent of the
number of hurricanes in any other calendar year.
Let N denote the number of hurricanes in a 20-year period.
(a) (15 points) Give possible values of N can take.
(b) (15 points) Give the probability mass function of N.
(c) (15 points) Calculate the probability that there are fewer than 3
hurricanes.
3. (40 points) Let X and Y be continuous random variables with joint density
function
╭ 24xy, for 0 < x < 1 and 0 < y < 1-x
f (x, y) = ╯
X, Y ╰ 0, otherwise
Calculate P(Y < X|X = 1/3).
4. (45 points) The joint density of (X, Y) is given by
╭ 3x, if 0 ≦ y ≦ x ≦ 1,
f(x, y) = ╯
╰ 0, otherwise
(a) (30 points) Compute the conditional density of Y given X = x.
(b) (15 points) Are X and Y independent? Justify your claim.
5. (30 points) Suppose X is uniform on (0, 1), Y is exponential with parameter
1, and X and Y are independent. Compute the PDF of X/Y.
6. (40 points) Suppose N is the number of flips of a fair coin until the first
2
head. Suppose Y is uniform on (0, N ). Suppose X is exponential with
＿
parameter √Y. Compute the expectation of X.
7. (40 points) A company agrees to accept the highest of four sealed bids on a
property. The four bids, X , X , X , and X , are regarded as four
1 2 3 4
independent random variables with common cumulative distribution function
1 3 5
F(x) =─(1 + sinπx) for─ ≦ x ≦─.
2 2 2
Calculate the expected value of the accepted bid.
8. (40 points) An insurance policy pays a total medical benefit consisting of
two parts for each claim. Let X represent the part of the benefit that is
paid to the surgeon, and let Y represent the part that is paid to the
hospital. The variance of X is 5,000, the variance of Y is 10,000, and the
variance of the total benefit, X+Y, is 17,000. Due to increasing medical
costs, the company that issues the policy decides to increase X by a flat
amount of 100 per claim and to increase Y by 10% per claim. Calculate the
variance of the total benefit after these revisions have been made.
9. (40 points) Assume a crime has been committed. It is known that the
perpetrator has certain characteristics, which occur with a small frequency
-8 8
p (say, 10 ) in a population of size n (say, 10 ). A person who matches
these characteristics has been found at random (e.g., at a routine traffic
stop or by airport security) and, since p is so small, charged with the
crime. There is no other evidence. What should the defense be?
(a) Let N be the number of people with given characteristics. Describe the
distribution of random variable N.
(b) Choose a person from among these N, label that person by C, the
criminal. Then, choose at random another person, A, who is arrested. The
question is whether C = A, that is, whether the arrested person is
guilty. Please determine P(C = A|N ≧ 1).
10. (40 points) Let X and Y be independent random variables. Express Var(XY) in
2 2
terms of [E(X)] , [E(Y)] , Var(X), and Var(Y).