170
Chapter 2 Discrete Random Variables and Probability Distributions
a. Let Y denote the number of calls answered in a one-minute period. Find the
probability distribution for Y .
b. Find E.Y .
Chapter Review
This chapter has considered several discrete probability distributions whose importance derives from the fact that they have various applications. Each of the distributions in
this chapter arises in one way or another from the binomial distribution. We began by defining a random variable as a real-valued function defined on the
points of a sample space. A typical example is throwing two dice and then recording the sum that appears. The sum is a random variable because it is a function, in this case
the sum, of the outcomes of the particular sample point that occurs. If X is a random variable, the probability distribution function, or pd f , is defined
as f .x = P.X = x:
A related function is the distribution function, is defined as F.x = P.X ≤ x:
The distribution function is not often used in this chapter, but has very important applications in the work to come.
Probability distributions are often distinguished and described by the values of their mean, ¼
x
; and their variance, ¦
2 x
: These are defined as
¼
x
= E.X = P
x
x · f .x and
¦
2 x
= Var.X = E.X − ¼
x 2
= P
x
. x − ¼
x 2
f .x; provided, of course, that the sums exist. The variance, ¦
2 x
; can also be calculated as
¦
2 x
= E.X
2
− [E.X]
2
: As a crude indication that ¦ actually measures the variation, or dispersion in a
random variable, we proved Tchebycheff’s Inequality: P.| X − ¼ | ≤ k · ¦ ≥ 1 −
1 k
2
where k is some positive quantity.
Chapter Review
171
We then turned to some specific discrete probability distributions. Of these, the single most important probability distribution is the binomial distribution, whose pdf
is given by
P. X = x = n
x p
x
q
n−x
; x = 0; 1; 2; : : : ; n where q = 1 − p:
This random variable arises from an experiment of n independent trials on each of which the result is one of two outcomes usually denoted by “success” or “failure”,
where p denotes the probability of success and X denotes the total number of successes. We used a recursion to find that, for the binomial distribution,
¼ = n · p
and ¦
2
= n · p · q: We then considered some statistical problems. We first considered the construction
of a confidence interval when sampling from a binomial distribution with known values of n and p. Frequently, however, p is unknown. We found an approximate 95
confidence interval for p to be
P n X + 2n − 2
p n
2
X + n
2
− n X
2
n
2
+ 4n ≤ p ≤
n X + 2n + 2 p
n
2
X + n
2
− n X
2
n
2
+ 4n = 0:95
where X is the observed number of successes in the binomial process with n trials. Tests of hypotheses were then considered. We examined tests of H
o
: p = p against the alternative H
a
: p = p
a
. The two types of error in testing a hypothesis are Þ
= probability of a Type I error = P[H
o
is rejected when it is true] and
þ = probability of a Type II error
= P[H
o
is accepted when it is false]. We considered the effect of the critical region – the set of observed values leading
to the rejection of the null hypothesis – on the size of þ and discussed 1 − þ; the power of the test. This is the probability that a false H
o
is correctly rejected. We derived the mean and the variance of a sample proportion arising from a sample
survey. Using these results we found that P
p
s
− 2 · r p · q
n ≤ p ≤ p
s
+ 2 · r p · q
n = 0:95
172
Chapter 2 Discrete Random Variables and Probability Distributions
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