4 P ROBABILITY D ISTRIBUTIONS
4.1 Random Variables
Variables 94 4.2 The Binomial
Distribution 98 4.3 The Hypergeometric
Distribution 103 4.4 The Mean and the
Variance of a Probability Distribution 107 4.5 Chebyshev’s
Theorem 114 4.6 The Poisson
Distribution and Rare Events 118
4.7 Poisson Processes 122 4.8 The Geometric and
Negative Binomial Distribution 124 4.9 The Multinomial
Distribution 127 4.10 Simulation 128
Review Exercises 132 Key Terms 133
I
n most statistical problems we are concerned with one number or a few numbers that are associated with the outcomes of experiments. When inspecting a manufac-tured product we may be interested only in the number of defectives; in the analysis of a road test we may be interested only in the average speed and the average fuel con-sumption; and in the study of the performance of a miniature rechargeable battery we may be interested only in its power and lifelength. All these numbers are associated with situations involving an element of chance—in other words, they are values of random variables.In the study of random variables we are usually interested in their probability distri-butions, namely, in the probabilities with which they take on the various values in their range. The introduction to random variables and probability distributions in Section 4.1 is followed by a discussion of various special probability distributions in Sections 4.2, 4.3, 4.6, 4.7, 4.8, and 4.9, and descriptions of the most important features of probability distributions in Sections 4.4 and 4.5.
4.1 Random Variables
To be more explicit about the concept of a random variable, let us refer again to the used car example of page 70 and the corresponding probabilities shown in Figure 3.9. Now let us refer to M1(low current mileage), P1(moderate price), and C1(inexpensive to operate) as preferred attributes. Suppose we are interested only in the number of preferred attributes a used car possesses. To find the probabilities that a used car will get 0, 1, 2, or 3 preferred attributes, let us refer to Figure 4.1, which is like Figure 3.9 in Chapter 3 except that we indicate for each outcome the number of preferred attributes. Adding the respective probabilities, we find that for 0 preferred attributes the probability is
0.07 + 0.06 + 0.03 + 0.02 = 0.18 and for one preferred attribute the probability is
0.01 + 0.01 + 0.16 + 0.10 + 0.02 + 0.05 + 0.14 + 0.01 = 0.50 For two preferred attributes, the probability is
0.06 + 0.07 + 0.02 + 0.09 + 0.05 = 0.29 and for three preferred attributes the probability is 0.03.
These results may be summarized, as in the following table, where x denotes a possible number of preferred attributes
x 0 1 2 3
Probability 0.18 0.50 0.29 0.03
94
Sec 4.1 Random Variables 95
Figure 4.1
Used cars and numbers of preferred attributes
The numbers 0, 1, 2, and 3 in this table are values of a random variable—the number of preferred attributes. Corresponding to each elementary outcome in the sample space there is exactly one value x for this random variable. That is, the ran-dom variable may be thought of as a function defined over the elements of the sample space. This is how we define random variables in general; they are functions defined over the elements of a sample space.
Random variables A random variable is any function that assigns a numerical value to each possible outcome.
The numerical value should be chosen to quantify an important characteristic of the outcome.
Random variables are denoted by capital letters X,Y, and so on, to distinguish them from their possible values given in lowercase x, y.
To find the probability that a random variable will take on any one value within its range, we proceed as in the above example. Indeed, the table which we obtained displays another function, called the probability distribution of the random vari-able. To denote the values of a probability distribution, we shall use such symbols as f (x), g(x),ϕ(y), h(z), and so on. Strictly speaking, it is the function f (x) = P(X = x) which assigns probability to each possible outcome x that is called the probability distribution. However, we will follow the common practice of also calling f(x) the probability distribution, with the understanding that we are referring to the function and that the range of x values is part of the definition.
Random variables are usually classified according to the number of values they can assume. In this chapter we shall limit our discussion to discrete random vari-ables, which can take on only a finite number, or a countable infinity of values;
continuous random variablesare taken up in Chapter 5.
Whenever possible, we try to express probability distributions by means of equations. Otherwise, we must give a table that actually exhibits the correspon-dence between the values of the random variable and the associated probabilities. For instance,
f (x)= 1
6 for x= 1, 2, 3, 4, 5, 6
gives the probability distribution for the number of points we roll with a balanced die.
Of course, not every function defined for the values of a random variable can serve as a probability distribution. Since the values of probability distributions are probabilities and one value of a random variable must always occur, it follows that if f (x) is a probability distribution, then
f (x)≥ 0 for all x and
all x
f (x)= 1
Probability distributions
The probability distribution of a discrete random variable X is a list of the possible values of X together with their probabilities
f (x)= P[X = x]
The probability distribution always satisfies the conditions f (x)≥ 0 and
all x
f (x)= 1
EXAMPLE 1 Checking for nonnegativity and total probability equals one Check whether the following can serve as probability distributions:
(a) f(x)= x− 2
2 for x= 1, 2, 3, 4 (b) h(x)= x2
25 for x= 0, 1, 2, 3, 4
Solution (a) This function cannot serve as a probability distribution because f (1) is negative.
(b) The function cannot serve as a probability distribution because the sum of the five probabilities is 6
5and not 1. j
It is often helpful to visualize probability distributions by means of graphs like those of Figure 4.2. The one on the left is called a probability histogram; the areas
Sec 4.1 Random Variables 97
Figure 4.2
Graphs of the probability distribution of the number of preferred attributes
0 1 2 3 x
f (x) 0.5
0 1 2 3 x
f (x)
Probability bar chart Probability histogram
0 1 2 3
0.4 0.3 0.2 0.1
0.5 0.4 0.3 0.2 0.1
of the rectangles are equal to the corresponding probabilities so their heights are proportional to the probabilities. The bases touch so that there are no gaps between the rectangles representing the successive values of the random variable. The one on the right is called a bar chart; the heights of the rectangles are also propor-tional to the corresponding probabilities, but they are narrow and their width is of no significance.
Besides the probability f (x) that the value of a random variable is x, there is an important related function. It gives the probability F (x) that the value of a random variable is less than or equal to x. Specifically,
F (x)= P[X ≤ x] for all −∞ < x < ∞
and we refer to the function F (x) as the cumulative distribution function or just the distribution function of the random variable. For any value x, it adds up, or accumulates, all the probability assigned to that value and smaller values.
Referring to the used car example and basing our calculations on the table on page 104, we get
x 0 1 2 3
F (x) 0.18 0.68 0.97 1.00
for the cumulative distribution function of the number of preferred attributes.
The cumulative distribution jumps the amount f ( x ) at x = 0, 1, 2, 3 and is constant between the values in the table as illustrated in Figure 4.3. The solid dots emphasize the fact that F (x) takes the upper value at jumps and this makes it con-tinuous from the right.
Figure 4.3
The cumulative distribution has jumps corresponding to
f (x)= P[X = x]
1.0
0 1 2 3
x F (x)
0.8 0.6 0.4 0.2
1.0
0 1 2 3
x f (x)
0.8 0.6 0.4 0.2