Applied Statistics and Probability for Engineers

5-2 MULTIPLE DISCRETE RANDOM VARIABLES 151

5-14. In the transmission of digital information, the probabil-

ity that a bit has high, moderate, and low distortion is 0.01, 0.10,

and 0.95, respectively. Suppose that three bits are transmitted

and that the amount of distortion of each bit is assumed to be in-

dependent. Let Xand Ydenote the number of bits with high and

moderate distortion out of the three, respectively. Determine

(a) (b)

(c) (d)

(e) (f ) Are Xand Yindependent?

5-15. A small-business Web site contains 100 pages and

60%, 30%, and 10% of the pages contain low, moderate, and

high graphic content, respectively. A sample of four pages is

selected without replacement, and Xand Ydenote the number

of pages with moderate and high graphics output in the

sample. Determine

(a) fXY 1 x, y 2 (b) fX 1 x 2

E 1 YƒX 12

E 1 X 2 fYƒ 11 y 2

fXY 1 x, y 2 fX 1 x 2

(c) (d)

(e) (f )

(g) Are Xand Yindependent?

5-16. A manufacturing company employs two inspecting

devices to sample a fraction of their output for quality control

purposes. The first inspection monitor is able to accurately

detect 99.3% of the defective items it receives, whereas the

second is able to do so in 99.7% of the cases. Assume that four

defective items are produced and sent out for inspection. Let X

and Ydenote the number of items that will be identified as

defective by inspecting devices 1 and 2, respectively. Assume

the devices are independent. Determine

(a) (b)

(c) (d)

(e) (f )

(g) Are Xand Yindependent?

E 1 YƒX 22 V 1 YƒX 22

E 1 X 2 fYƒ 2 1 y 2

fXY 1 x, y 2 fX 1 x 2

E 1 Y 0 X 32 V 1 Y 0 X 32

E 1 X 2 fYƒ 3 1 y 2

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 1 , X 2 , X 3 , and X 4 denote the

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 1 ,x 2 ,x 3 , and x 4 such that x 1 x 2 x 3  x 4 20 receive positive probability in the prob-

ability distribution.

In general, given discrete random variables the joint probability dis-

tribution of is a description of the set of points in the

range of along with the probability of each point. A joint probability mass

function is a simple extension of a bivariate probability mass function.

X 1 , X 2 , X 3 ,p, Xp,

X 1 , X 2 , X 3 ,p, Xp 1 x 1 , x 2 , x 3 ,p, xp 2

X 1 , X 2 , X 3 ,p, Xp,

The joint probability mass function of is

(5-8)

for all points 1 x 1 , x 2 ,p, xp 2 in the range of X 1 , X 2 ,p, Xp.

fX 1 X 2 p (^) Xp 1 x 1 , x 2 ,p, xp 2 P 1 X 1 x 1 , X 2 x 2 ,p, Xpxp 2
X 1 , X 2 ,p, Xp
Definition
A marginal probability distribution is a simple extension of the result for two random
variables.
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