課程名稱︰機率
課程性質︰大二下必修
課程教師︰洪一平
開課系所︰資工系
考試時間︰2006/03/30 14:20~17:20
試題 :
1. (10%) Suppose that P(A) = 0.7, P(B) = 0.5, and P([A [ B]0) = 0.1.
(a) Find P(A∩B).
(b) Give P(A|B).
(c) Give P(B|A).
2. (10%) Let P(A) = 0.3 and P(B) = 0.6.
(a) Find P(A∪B) when A and B are independent.
(b) Find P(A|B) when A and B are mutually exclusive.
3. (10%) Software to detect fraud in consumer phone cards tracks the number
of metropolitaion areas where calls originate each day. It is found that
1% of the legitimate users originate calls from two or more metropolitan
areas in a single day. However, 30% of fraudulent users originate calls
from two or more metropolitan areas in a single day. The proportion of
fraudulent users is 0.01%. If the same user originates calls from two or
more metropolitan areas in a single day, what is the probability that
the user is fraudulent?
4. (10%) Let X be the number of accidents in a factory per week having
p.m.f.
1
f(x) = -------------- , x = 0, 1, 2, · ·
(x + 1)(x + 2)
Find the conditional probability of X≧4, given that X≧1.
5. (10%) Let the random variable X be the number of days that a certain
patient needs to be in the hospital. Say X has the p.m.f.
5-x
f(x) = ------ , x = 1, 2, 3, 4.
10
If the patient is to receive from an insurance company $200 for each of
the first two days in the hospital and $100 for each day after the first
two days, what is the expected payment for the hospitalization?
6. (10%) Suppose that X has a discrete uniform distribution on the integers
0 through 9. Determine the mean, variance, and standard deviation of the
random variable Y = 5X and compare to the corresponding results for X.
7. (10%) A recent national study showed that approximately 45% of college
students binge drink. Let X equal the number of students in a random
sample of size n = 12 who binge drink. Find the probability that
(a) X is at most 5.
(b) X is at least 6.
(c) X is equal to 7.
(d) Give the mean, variance, and standard deviation of X.
8. (10%) Samples of 20 parts from a metal punching process are selected
every hour. Typically, 1% of the parts require rework. Let X denote the
number of parts in the sample of 20 that require rework. A proces problem
is suspected if X exceeds its mean by more than three standard
devariaitions.
(a) If the percentage of parts that require rework remains at 1%, what
is the probability that X exceeds its mean by more than three
standard deviations?
(b) If the rework percentage increases to 4%, what is the probability
that X exceeds 1?
(c) If the rework percentage increases to 4%, what is the probability
that X exceeds 1 in at least one of the next five hours of samples?
9. (10%) Let the moment-generating function M(t) of X exist for h < t < h.
Consider the function R(t) = lnM(t). The first two derivatives of R(t)
are, respectively,
M'(t) M(t)M''(t) - [M'(t)]^2
R'(t) = ------- , and R''(t) = -------------------------
M(t) [M(t)]^2
Setting t = 0, show that
(a) μ = R'(0).
(b) σ^2 = R''(0).
(10%) Let X have a Poisson distribution with a mean of 4. Find
(a) P(2≦X≦5)
(b) P(X≧5)
(c) P(X≦3)
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