Applied Statistics and Probability for Engineers

66 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3-29. Determine the cumulative distribution function for

the random variable in Exercise 3-19.

3-30. Determine the cumulative distribution function for

the random variable in Exercise 3-20.

3-31. Determine the cumulative distribution function for

the random variable in Exercise 3-22.

3-32. Determine the cumulative distribution function for

the variable in Exercise 3-23.

Verify that the following functions are cumulative distribution

functions, and determine the probability mass function and the

requested probabilities.

3-33.

(a) (b)

(c) (d)

3-34. Errors in an experimental transmission channel are

found when the transmission is checked by a certifier that de-

tects missing pulses. The number of errors found in an eight-

bit byte is a random variable with the following distribution:

F 1 x 2 μ

0 x 1

0.7 1 x 4

0.9 4 x 7

17 x

P 11 X 22 P 1 X 22

P 1 X 32 P 1 X 22

F 1 x 2 •

0 x 1

0.5 1 x 3

13 x

3-4 MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE

Two numbers are often used to summarize a probability distribution for a random variable X.

The mean is a measure of the center or middle of the probability distribution, and the variance

is a measure of the dispersion, or variability in the distribution. These two measures do not

uniquely identify a probability distribution. That is, two different distributions can have the

same mean and variance. Still, these measures are simple, useful summaries of the probabil-

ity distribution of X.

The mean or expected value of the discrete random variable X, denoted as or is

(3-3)

The varianceof X, denoted as or is

The standard deviationof Xis . 22

2 V 1 X 2 E 1 X 22  a

x

1 x 22 f 1 x 2  a

x

x^2 f 1 x 2 ^2

2 V 1 X 2 ,

E 1 X 2  a

x

xf 1 x 2

E 1 X 2 ,

Definition

Determine each of the following probabilities:

(a) (b)

(c) (d)

(e)

3-35.

(a) (b)

(c) (d)

(e) (f)

3-36. The thickness of wood paneling (in inches) that a cus-

tomer orders is a random variable with the following cumula-

tive distribution function:

Determine the following probabilities:

(a) (b)

(c) (d)

(e)P 1 X (^1)  22
P 1 X (^5)  162 P 1 X (^1)  42
P 1 X (^1)  182 P 1 X (^1)  42
F 1 x 2 μ
0 x (^1)  8
0.2 (^1)  8 x (^1)  4
0.9 (^1)  4 x (^3)  8
(^13)  8 x
P 10 X 102 P 1  10 X 102
P 140 X 602 P 1 X 02
P 1 X 502 P 1 X 402
F 1 x 2 μ
0 x 10
0.25  10 x 30
0.75 30 x 50
150 x
P 1 X 22
P 1 X 52 P 1 X 42
P 1 X 42 P 1 X 72
PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 66