Applied Statistics and Probability for Engineers

144 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

and

The probabilities for all points in Fig. 5-1 are shown next to the point and the figure describes

the joint probability distribution of Xand Y.

5-1.2 Marginal Probability Distributions

If more than one random variable is defined in a random experiment, it is important to distin-

guish between the joint probability distribution of Xand Yand the probability distribution of

each variable individually. The individual probability distribution of a random variable is re-

ferred to as itsmarginal probability distribution.In Example 5-1, we mentioned that the

marginal probability distribution of Xis binomial with n4 and p0.9 and the marginal

probability distribution of Yis binomial with n4 and p0.08.

In general, the marginal probability distribution of Xcan be determined from the joint

probability distribution of Xand other random variables. For example, to determine P(Xx),

we sum P(X x, Yy) over all points in the range of (X, Y) for which Xx. Subscripts on

the probability mass functions distinguish between the random variables.

EXAMPLE 5-3 The joint probability distribution of Xand Yin Fig. 5-1 can be used to find the marginal prob-

ability distribution of X. For example,

As expected, this probability matches the result obtained from the binomial probability distribu-

tion for X; that is,. The marginal probability distribution for X

is found by summing the probabilities in each column, whereas the marginal probability distribu-

tion for Yis found by summing the probabilities in each row. The results are shown in Fig. 5-2.

Although the marginal probability distribution of Xin the previous example can be

determined directly from the description of the experiment, in some problems the marginal

probability distribution is determined from the joint probability distribution.

P 1 X 32  143 2 0.9^3 0.1^1 0.292

0.05830.23330.292

P 1 X 32 P 1 X3, Y 02 P 1 X3, Y 12

fXY 1 2, 1 2 P 1 X2, Y 12 0.0156

fX(x) =0.0001 0.0036 0.0486 0.2916 0.6561

0.71637

0.24925

0.03250

0.00188

0.00004

fY(y) =

x

y

0 1 2 3 4

0

1

2

3

4

Figure 5-2 Marginal

probability distribu-

tions ofXandYfrom

Fig. 5-1.

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