课程名称︰机率导论
课程性质︰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）︰2013/05/09
考试时限（分钟）：13:20PM-15:10
试题 :
Introductory Probability 机率导论
期中考特考
考试时间: 13:20PM-15:10 Thursday, May 9th, 2013
共215分，最高得分为180分。解答过程须详列。
当题目中所提供的参考答案，都与你的答案不同时，请注明以上皆非。
1. (points) Let X and Y denote the values of two stocks at the end of a
five-year period. X is uniformly distributed on the interval (0,12). Given
X = x, Y is uniformly distributed on the interval (0,x).
(a) (15 points) Derive joint density function of (X,Y).
(b) (15 points) Determine Cov(X,Y) according to this model. 参考答案为(A)0(B)4
(C)6(D)12(E)24。
2. (20 points) A device contains 两个线路板. 第二个线路板 is a backup for the
first, 所以只当第一个线路板异常时，第二个线路板才会启动。 故the device fails
仅当第二个线路板也异常。 令 X 及 Y 分别为第一个线路板的异常时间及第二个线路
板的异常时间， X and Y have joint probability density function
╭
│6e^-x ˙ e^-2y for 0 < x < y < ∞
f(x,y) = ﹤
│0 otherwise
╰
What is the expected time at which the device fails?
参考答案为(A)0.33(B)0.50(C)0.67(D)0.83(E)1.50。
3. (20 points) A device that continuously measures and and records 地震活动 is
placed in a remote region. The time, T, to failure of this device is
exponentially distributed with mean 3 years. Since the device will not be
monitored during its first two years of service, the time to discovery of
its failure is X = max(T,2). Determine E[X].
参考答案为(A) 2+(1/3)exp(-6) (B) 2-2exp(-2/3 + 5exp(-4/3) (C)3
(D)2 + 3exp(-2/3) (E)5。
4. (30 points) A car dealership sells 0, 1, or 2 luxury cars on any day. When
selling a car, the dealer also tries to persuade the customer to buy an
extened warranty for the car. Let X denote the number of luxury cars sold in
a given day, and let Y denote the number of extended warranties sold, and
suppose that
╭1/6 for (x,y) = (0,0),
│1/12 for (x,y) = (1,0),
│1/6 for (x,y) = (1,1),
P(X = x, Y = y) = ﹤1/12 for (x,y) = (2,0),
│1/3 for (x,y) = (2,1),
│1/6 for (x,y) = (2,2).
╰
What is the variance of X? 参考答案为 (A)0.47(B)0.58(C)0.83(D)1.42(E)2.58
5. (20 points) A device contains two components. The device fails if either
component fails. The joint density function of the life times of the
components, measured in hours, is f(s,t), where 0<s<1 and 0<t<1. What is the
probability that the device fails during the first half hour of operation?
写出其数学表达式。
6. (30 points) The stock prices of two companies at the end of any given year
are modeled with random variables X and Y that follow a distribution with
joint density function
╭
│ 2x for 0 < x < 1, x < y < x+1,
f(x,y) = ﹤
│ 0 otherwise.
╰
What is the conditional variance of Y given that X = x?
(A)1/12(B)7/6(C)x + 1/2(D)x^2 - 1/6(E)x^2 + x + 1/3
7. (30 points) A company prices its hurricane insurance, 飓风保险, using 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.
在该公司的假设条件之下，
(a) (15 points) Calculate the probability that there are fewer than 3
hurricanes in a 20-year period.
(b) (15 points) Give an approximation of (a) in terms of a Poisson random
variable with mean λ.
8. (15 points) Let C_1, C_2, and C_3 be independent events with probability
c c
1/2, 1/3, 1/4, respectively. Compute P(C_1∩C_2|C_3∪C_2).
9. (20 points) Let X be a Poisson random variable with parameter λ. Find a
simple expression of E[1/(1+X)].