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The midterm Exam of Probability & Statistics (I)
Department of Information Management
Date: 8 November 2004
09:30~11:30
NOTICE!
1 You can use the calculator for some complicated computations. However,
the Notebook computers, PDAs, Smart phones, or anything computable
else will not be allowed.
2 Answer the following questions in the closed-book context and honesty.
3 Any evidence of cheating will result in a failing grade for the course and
further disciplinary action through the appropriate Chi Nan procedures and
committees.
I. Short answer
1. Explain the purpose of defining a random variable. (5%)
2. Compare the probability mass function and the probability density function.
(5%)
3. A competitive campaign is coming. A candidate’s manager wants to use the
member list of one famous Internet club as a sampling frame in order to
investigate the vote-getting rate. Please comment on this approach. (5%)
4.
5.
Compare between the two mutual-exclusive events and two independent events.
(5%)
Please explain why the variance Var(X+X) is equal to 4Var(X) rather than
2Var(X)? (5%)
II. Computation problem
1 Please rearrange the following data set into the form of stem-and-leaf plotting.
(5%)
17,25,83,62,54,59,43,41,58,19,26,72,77,65,53,47,51,32,34,51,56,23
2
3
Following the sorted data set in the above question, please draw a
box-and-whisker plot. Besides, you have to notify the median, the first quartile,
and the third quartile value, respectively, in the plot. (10%)
The joint density function of (X,Y) is given by
 xe(  x  y ) , x  0, y  0
f ( x, y)  
0, otherwise
a
Compute the marginal p.d.f. of X (10%)
b
4
5
6
Compute the marginal p.d.f. of Y (10%)
c
Are X and Y mutually independent? (5%)
Suppose that an insurance company classifies people into one of three
classes—good risks, average risks, and bad risks. Their records indicate that the
probabilities that good, average, and bad risk persons will be involved in an
accident over a 1-year span are, respectively, 0.05, 0.15, and 0.30. If 20 percent of
the populations are “good risks,” 50 percent are “average risks,” and 30 percent
are “bad risks,” what proportions of people have accidents in a fixed year? (5%)
If someone had no accidents in 2003, what is the probability that he or she is a
good risk? (10%)
If E[X]=2 and E[X2]=8, calculate (a) E[(2+4X)2] (5%) and (b) E[X2+(X+1)2]
(5%)
Suppose that the closing probability of every relay is given by p and all relays
function independently. Please compute the probability that a current flows
between A and B for the following two circuits respectively. (10%)
A
B
(a)
7
A
B
(b)
The Christmas party is coming soon. One member of university students
association proposed an attractive marketing incentive by giving luxury gifts to
any two having the same birthday. Now suppose that you are the chair of student
association. Would you accept this proposal? Why? (5%)