Applied Statistics and Probability for Engineers

5-2 MULTIPLE DISCRETE RANDOM VARIABLES 155

The multinomial distribution is considered a multivariable extension of the binomial

distribution.

EXAMPLE 5-13 In Example 5-12, 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. The probability that 12 of the bits

received are E, 6 are G, 2 are F, and 0 are Pis

Each trial in a multinomial random experiment can be regarded as either generating or not

generating a result in class i, for each i1, 2,... , k. Because the random variable Xiis the

number of trials that result in class i, Xihas a binomial distribution.

P 1 X 1 12, X 2 6, X 3 2, X 4  02 

20!

12! 6! 2! 0!

0.6^12 0.3^6 0.08^2 0.02^0 0.0358

If X 1 , X 2 ,... , Xkhave a multinomial distribution, the marginal probability distribu-

tion of Xiis binomial with

E 1 Xi 2 npi and V 1 Xi 2 npi 11 pi 2 (5-14)

EXAMPLE 5-14 In Example 5-13, the marginal probability distribution of X 2 is binomial with n20 and

p0.3. Furthermore, the joint marginal probability distribution of X 2 and X 3 is found as

follows. The P(X 2 x 2 , X 3 x 3 ) is the probability that exactly x 2 trials result in Gand that x 3

result in F. The remaining nx 2 x 3 trials must result in either Eor P. Consequently, we can

consider each trial in the experiment to result in one of three classes, {G}, {F}, or {E,P}, with

probabilities 0.3, 0.08, and 0.60.020.62, respectively. With these new classes, we can

consider the trials to comprise a new multinomial experiment. Therefore,

The joint probability distribution of other sets of variables can be found similarly.

EXERCISES FOR SECTION 5-2



n!

x 2 !x 3! 1 nx 2 x 32!

1 0.3 2 x^21 0.08 2 x^31 0.62 2 nx^2 x^3

fX 2 X 3 1 x 2 , x 32 P 1 X 2 x 2 , X 3 x 32

5-17. Suppose the random variables X, Y, and Zhave the

following joint probability distribution

xyz f(x, y, z)

1 1 1 0.05

1 1 2 0.10

1 2 1 0.15

1 2 2 0.20

2 1 1 0.20

2 1 2 0.15

2 2 1 0.10

2 2 2 0.05

Determine the following:

(a) (b)

(c) (d)

(e)

5-18. Continuation of Exercise 5-17. Determine the follow-

ing:

(a) ( b )

(c)

5-19. Continuation of Exercise 5-17. Determine the condi-

tional probability distribution of Xgiven that Y1 and Z2.

5-20. Based on the number of voids, a ferrite slab is classi-

fied as either high, medium, or low. Historically, 5% of the

slabs are classified as high, 85% as medium, and 10% as low.

P 1 X 1 ƒY1, Z 22

P 1 X 1 ƒY 12 P 1 X1, Y 1 ƒZ 22

E 1 X 2

P 1 Z1.5 2 P 1 X 1 or Z 22

P 1 X 22 P 1 X1, Y 22

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