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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 X denote the length of one dimenson of an injection-molded part, and let the
continuous random variable Y denote 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 X and Y in specific regions of two-dimensional space. For example, a small pressure in-
crease might generate parts such that both X and Y are greater than their respective targets and
a small pressure decrease might generate parts such that X and Y are both less than their re-
spective targets. Therefore, based on pressure variations, we expect that the probability of a
part with X much greater than its target and Y much less than its target is small. Knowledge of
the joint probability distribution of X and Y provides information that is not obvious from the
marginal probability distributions.
The joint probability distribution of two continuous random variables X and Y can be
specified by providing a method for calculating the probability that X and Y assume a value in
any region R of two-dimensional space. Analogous to the probability density function of a sin-
gle continuous random variable, a joint probability density function can be defined over
two-dimensional space. The double integral of f 1x, y2 over a region R provides the proba-
XY
bility that 1X, Y 2 assumes a value in R. This integral can be interpreted as the volume under the
surface f XY 1x, y2 over the region R.
A joint probability density function for X and Y is shown in Fig. 5-6. The probability
that 1X, Y 2 assumes a value in the region R equals 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 X and Y.

Definition
A joint probability density function for the continuous random variables X and Y,
1x, y2, satisfies the following properties:
denoted as f XY
(1) f XY 1x, y2 0 for all x, y

(2) f XY 1x, y2 dx dy 1

(3) For any region R of two-dimensional space

P13X, Y4 R2 f 1x, y2 dx dy (5-15)
XY

R

Typically, f XY 1x, y2 is defined over all of two-dimensional space by assuming that
f XY 1x, y2 0 for all points for which f XY 1x, y2 is not specified.