Page 202 - Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 185

PROBLEMS
6.1 The random variable X has a binomial distribution with parameters (n, p). Using the
formulation given by Equation (6.10), derive its probability mass function (pmf),
mean, and variance and compare them with results given in Equations (6.2) and
(6.8). (Hint: see Example 4.18, page 109).
6.2 Let X be the number of defective parts produced on a certain production line. It is
known that, for a given lot, X is binomial, with mean equal to 240, and variance
48. Determine the pmf of X and the probability that none of the parts is defective
in this lot.
6.3 An experiment is repeated 5 times. Assuming that the probability of an experiment
being successful is 0.75 and assuming independence of experimental outcomes:
(a) What is the probability that all five experiments will be successful?
(b) How many experiments are expected to succeed on average?
6.4 Suppose that the probability is 0.2 that the air pollution level in a given region will
be in the unsafe range. What is the probability that the level will be unsafe 7 days in
a 30-day month? What is the average number of ‘unsafe’ days in a 30-day month?
6.5 An airline estimates that 5% of the people making reservations on a certain flight
will not show up. Consequently, their policy is to sell 84 tickets for a flight that can
only hold 80 passengers. What is the probability that there will be a seat available
for every passenger that shows up? What is the average number of no-shows?
6.6 Assuming that each child has probability 0.51 of being a boy:
(a) Find the probability that a family of four children will have (i) exactly one boy,
(ii) exactly one girl, (iii) at least one boy, and (iv) at least one girl.
(b) Find the number of children a couple should have in order that the probability
of their having at least two boys will be greater than 0.75.
6.7 Suppose there are five customers served by a telephone exchange and that each
customer may demand one line or none in any given minute. The probability of
demanding one line is 0.25 for each customer, and the demands are independent.
(a) What is the probability distribution function of X, a random variable repre-
senting the number of lines required in any given minute?
(b) If the exchange has three lines, what is the probability that the customers will
all be satisfied?
6.8 A park-by-permit-only facility has m parking spaces. A total of n (n m) parking
permits are issued, and each permit holder has a probability p of using the facility
in a given period.
(a) Determine the probability that a permit holder will be denied a parking space
in the given time period.
(b) Determine the expected number of people turned away in the given time period.
6.9 For the hypergeometric distribution given by Equation (6.13), show that as n !1
it approaches the binomial distribution with parameters m and n 1 /n; that is,
k
m n 1 n 1
m k
p k†ˆ 1 ; k ˆ 0; 1; ... ; m:
Z
k n n
and thus that the hypergeometric distribution can be approximated by a binomial
distribution as n !1.

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