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Chapter 2 Discrete Random Variables and Probability Distributions
substance. Approximate the probability that a particular one of the measurements is within 5¦=4 units of ¼:
10. A manufacturer ships parts in lots of 1000 and makes a profit of 50 per lot sold.
The purchaser, however, subjects the product to a sampling inspection plan as follows: 10 parts are selected at random. If none of these parts is defective, the lot
is purchased; if one part is defective, the manufacturer returns 10 to the buyer; if 2 or more parts are found to be defective, the entire lot is returned at a net loss of
25 to the manufacturer. What is the manufacturer’s expected profit if 10 of the parts are defective? Assume that the sampling is done with replacement.
11. In a lot of 6 batteries, one is worn out. A technician tests the batteries one at a time
until the worn-out battery is found. Tested batteries are put aside, but after every third test the tester takes a break and another worker, unaware of the test, returns
one of the tested batteries to the set of batteries not yet tested.
a. Find the probability distribution for X, the number of tests required to identify
the worn-out battery.
b. Assume the first test of each set of 3 tests costs 5 and that each of the next 2
tests in each set of three tests costs 2. Find the increase in the expected cost of locating the worn-out battery due to the unaware worker.
12. A carnival game consists of hitting a lever with a sledge hammer to propel a weight
upward toward a bell. Because the hammer is quite heavy, the chance of ringing the bell declines with the number of attempts; in particular, the probability of ringing
the bell on the ith attempt is
3 4
i
. For a fee, the carnival sells you the privilege of swinging the hammer until the bell rings or until you have made 3 attempts,
whichever occurs first.
a. Find the probability distribution of X, the number of hits taken. b. The prize for ringing the bell on the ith try is .4 − i, i = 1; 2; 3. How much
should the carnival charge for playing the game if it wants an expected profit of 1 per customer?
13. Suppose X is a random variable defined on the points x = 0, 1, 2, 3,: : : Calculate
∞
X
x=0
P. X x: There are many very important specific discrete probability distribution functions
that arise in practical applications. Having established some general properties, we now turn to discussions of several of the most important of these distributions.
Occasionally random variables in apparently different situations actually arise from common assumptions and hence lead to the same probability distribution function. We
now investigate some of these special circumstances and the probability distribution functions that result.
2.4 Binomial Distribution
99
2.4
c
Binomial Distribution
Among all discrete probability distribution functions, the most commonly occurring one, arising in a great variety of applications, is called the binomial probability distri-
bution function. Consider an experiment where, on each trial of the experiment, one of only two
outcomes occurs, which we describe as success, S or failure, F. For example, a manufactured part is either good or does not meet specifications; a student’s examina-
tion score is passing or it is not; a team wins a basketball game or it does not – these are some examples, and the reader can no doubt think of many more. One of these
outcomes can be associated with success and the other with failure; it does not matter which is which.
In addition to the restriction that there be two and only two outcomes on each trial of the experiment, suppose further that the trials are independent, and that the
probabilities of success or failure at each trial remain constant from trial to trial and do not change with subsequent performances of the experiment.
The individual trials of such an experiment are often called Bernoulli trials. Consider, as a specific example, five independent trials with probability
2 3
of success at any trial. Then, if interest centers on the occurrence of exactly three successes,
we note that exactly three successes can occur in ten different ways: S S S F F; S S F S F; S F S S F; F S S S F; S F S F S;
S S F F S; F S S F S; S F F S S; F S F S S; F F S S S: There are
5 3
Ð = 10 of these mutually exclusive orders. Each has probability
2 3
3
·
1 3
2
, so
P .exactly 3 S
′
s in 5 trials = 5
3 ·
2 3
3
· 1
3
2
= 80
243 :
Now return to the general situation. Let the probabilities be P.S = p and P. F = q = 1 − p, and let the random variable X denote the number of successes
in n trials of the experiment. Any specific sequence of exactly x successes and n − x failures has probability p
x
· q
n−x
: The successes in such a sequence can occur at
n x
Ð positions so, since the sequences are mutually exclusive,
P. X = x = n
x p
x
· q
n−x
; x = 0; 1; 2; : : : ; n;
2.3 giving the probability distribution function for a binomial random variable.
Although the binomial random variable occurs in many different situations, a perfect model for any binomial situation is that of observing the number of heads when
a coin loaded so that the probability of a head is p and that of a tail is q = 1 − p is tossed n times.
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Chapter 2 Discrete Random Variables and Probability Distributions
