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5-2 MULTIPLE DISCRETE RANDOM VARIABLES 151

5-14. In the transmission of digital information, the probabil- (c) E1X 2 (d) f Yƒ3 1 y2
ity that a bit has high, moderate, and low distortion is 0.01, 0.10, (e) E1Y 0 X 32 (f) V1Y 0 X 32
and 0.95, respectively. Suppose that three bits are transmitted (g) Are X and Y independent?
and that the amount of distortion of each bit is assumed to be in- 5-16. A manufacturing company employs two inspecting
dependent. Let X and Y denote the number of bits with high and devices to sample a fraction of their output for quality control
moderate distortion out of the three, respectively. Determine purposes. The first inspection monitor is able to accurately
(a) f XY 1x, y2 (b) f X 1x2 detect 99.3% of the defective items it receives, whereas the
(c) E1X 2 (d) f Y ƒ1 1 y2 second is able to do so in 99.7% of the cases. Assume that four
(e) E1Y ƒ X 12 (f) Are X and Y independent? defective items are produced and sent out for inspection. Let X
5-15. A small-business Web site contains 100 pages and and Y denote the number of items that will be identified as
60%, 30%, and 10% of the pages contain low, moderate, and defective by inspecting devices 1 and 2, respectively. Assume
high graphic content, respectively. A sample of four pages is the devices are independent. Determine
selected without replacement, and X and Y denote the number (a) f XY 1x, y 2 (b) f X 1x2
of pages with moderate and high graphics output in the (c) E1X 2 (d) f Yƒ2 1y2
sample. Determine (e) E1Y ƒ X 22 (f) V1Y ƒ X 22
(a) f XY 1x, y2 (b) f X 1x2 (g) Are X and Y independent?

5-2 MULTIPLE DISCRETE RANDOM VARIABLES

5-2.1 Joint Probability Distributions

EXAMPLE 5-10 In some cases, more than two random variables are defined in a random experiment, and
the concepts presented earlier in the chapter can easily be extended. The notation can be
cumbersome and if doubts arise, it is helpful to refer to the equivalent concept for two ran-
dom variables. Suppose that the quality of each bit received in Example 5-1 is categorized
even more finely into one of the four classes, excellent, good, fair, or poor, denoted by
E, G, F, and P, respectively. Also, let the random variables X , X , X , and X denote the
3
4
1
2
number of bits that are E, G, F, and P, respectively, in a transmission of 20 bits. In this
example, we are interested in the joint probability distribution of four random variables.
Because each of the 20 bits is categorized into one of the four classes, only values for
x , x , x , and x such that x x x x 20 receive positive probability in the prob-
1
3
4
4
3
2
1
2
ability distribution.
In general, given discrete random variables X , X , X , p , X , the joint probability dis-
3
1
2
p
is a description of the set of points 1x 1 , x 2 , x 3 , p , x p 2 in the
tribution of X 1 , X 2 , X 3 , p , X p
range of X 1 , X 2 , X 3 , p , X p , along with the probability of each point. A joint probability mass
function is a simple extension of a bivariate probability mass function.
Definition
The joint probability mass function of X , X , p , X p is
2
1
1
2
2
p
p
1x , x , p , x 2 P1X x , X x , p , X x 2 (5-8)
1
1
2
p
f X 1 X 2 p X p
for all points 1x 1 , x 2 , p , x p 2 in the range of X , X 2 , p , X p .
1
A marginal probability distribution is a simple extension of the result for two random
variables.