Applied Statistics and Probability for Engineers

5-3 TWO CONTINUOUS RANDOM VARIABLES 157

5-3 TWO CONTINUOUS RANDOM VARIABLES

5-3.1 Joint Probability Distributions

Our presentation of the joint probability distribution of two continuous random variables is

similar to our discussion of two discrete random variables. As an example, let the continuous

random variable Xdenote the length of one dimenson of an injection-molded part, and let the

continuous random variable Ydenote the length of another dimension. The sample space of

the random experiment consists of points in two dimensions.

We can study each random variable separately. However, because the two random vari-

ables are measurements from the same part, small disturbances in the injection-molding

process, such as pressure and temperature variations, might be more likely to generate values

for Xand Yin specific regions of two-dimensional space. For example, a small pressure in-

crease might generate parts such that both Xand Yare greater than their respective targets and

a small pressure decrease might generate parts such that Xand Yare both less than their re-

spective targets. Therefore, based on pressure variations, we expect that the probability of a

part with Xmuch greater than its target and Ymuch less than its target is small. Knowledge of

the joint probability distribution of Xand Yprovides information that is not obvious from the

marginal probability distributions.

The joint probability distribution of two continuous random variables Xand Ycan be

specified by providing a method for calculating the probability that Xand Yassume a value in

any region Rof two-dimensional space. Analogous to the probability density function of a sin-

gle continuous random variable, a joint probability density functioncan be defined over

two-dimensional space. The double integral of over a region Rprovides the proba-

bility that assumes a value in R. This integral can be interpreted as the volume under the

surface over the region R.

A joint probability density function for Xand Yis shown in Fig. 5-6. The probability

that assumes a value in the region Requals the volume of the shaded region in

Fig. 5-6. In this manner, a joint probability density function is used to determine probabil-

ities for Xand Y.

1 X, Y 2

fXY 1 x, y 2

1 X, Y 2

fXY 1 x, y 2

A joint probability density functionfor the continuous random variables Xand Y,

denoted as satisfies the following properties:

(1)

(2)

(3) For any region Rof two-dimensional space

P 13 X, Y 4 R 2  (5-15)

R

fXY 1 x, y 2 dx dy





(^) 

fXY 1 x, y 2 dx dy 1
fXY 1 x, y 2 0 for all x, y
fXY 1 x, y 2 ,
Definition
Typically, is defined over all of two-dimensional space by assuming that
fXY 1 x, y 2  0 for all points for which fXY 1 x, y 2 is not specified.
fXY 1 x, y 2
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