Page 178 - Applied statistics and probability for engineers

156 Chapter 5/Joint Probability Distributions

Learning Objectives
After careful study of this chapter, you should be able to do the following:

1. Use joint probability mass functions and joint probability density functions to calculate probabilities
2. Calculate marginal and conditional probability distributions from joint probability distributions
3. Interpret and calculate covariances and correlations between random variables
4. Use the multinomial distribution to determine probabilities
5. Understand properties of a bivariate normal distribution and be able to draw contour plots for the
probability density function
6. Calculate means and variances for linear combinations of random variables and calculate
probabilities for linear combinations of normally distributed random variables
7. Determine the distribution of a general function of a random variable
8. Calculate moment generating functions and use the functions to determine moments and distributions

In Chapters 3 and 4, you studied probability distributions for a single random variable. How-
ever, it is often useful to have more than one random variable deined in a random experi-

ment. For example, in the classiication of transmitted and received signals, each signal can be

classiied as high, medium, or low quality. We might deine the random variable X to be the

number of high-quality signals received and the random variable Y to be the number of low-
quality signals received. In another example, the continuous random variable X can denote the
length of one dimension of an injection-molded part, and the continuous random variable Y
might denote the length of another dimension. We might be interested in probabilities that can
be expressed in terms of both X and Y. For example, if the specii cations for X and Y are (2.95
to 3.05) and (7.60 to 7.80) millimeters, respectively, we might be interested in the probability
(
that a part satisies both speciications; that is, P 2.95 , X , 3.05 and 7.60 , , 7.80).
Y

Because the two random variables 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 speciic regions of two-dimensional space. For exam-
ple, a small pressure increase 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 respective 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.
In general, if X and Y are two random variables, the probability distribution that dei nes
their simultaneous behavior is called a joint probability distribution. In this chapter, we
investigate some important properties of these joint distributions.

5-1 Two or More Random Variables

5-1.1 JOINT PROBABILITY DISTRIBUTIONS

Joint Probability Mass Function
For simplicity, we begin by considering random experiments in which only two random vari-
ables are studied. In later sections, we generalize the presentation to the joint probability
distribution of more than two random variables.

Example 5-1 Mobile Response Time The response time is the speed of page downloads and it is critical for
a mobile Web site. As the response time increases, customers become more frustrated and poten-
tially abandon the site for a competitive one. Let X denote the number of bars of service, and let Y denote the response
time (to the nearest second) for a particular user and site.