Applied Statistics and Probability for Engineers

108 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

EXAMPLE 4-9 Let the continuous random variable Xdenote the current measured in a thin copper wire in

milliamperes. Assume that the range of Xis [0, 20 mA], and assume that the probability den-

sity function of Xis

What is the probability that a measurement of current is between 5 and 10 milliamperes?

The requested probability is shown as the shaded area in Fig. 4-9.

The mean and variance formulas can be applied with and Therefore,

Consequently, the standard deviation of Xis 5.77 mA.

The cumulative distribution function of a continuous uniform random variable is ob-

tained by integration. If

Therefore, the complete description of the cumulative distribution function of a continuous

uniform random variable is

An example of F(x) for a continuous uniform random variable is shown in Fig. 4-6.

EXERCISES FOR SECTION 4-5

F 1 x 2 •

0 xa

1 xa 2 1 ba 2 axb

1 bx

F 1 x 2 

x

a

1 1 ba 2 dux 1 ba 2 a 1 ba 2

axb,

E 1 X 2 10 mA and V 1 X 2  202 12 33.33 mA^2

a 0 b20.

 51 0.05 2 0.25

P 15 X 102 

10

5

f 1 x 2 dx

f 1 x 2 0.05, 0x20.

Figure 4-9 Probability for Example 4-9.

x

f(x)

0 5 10 15 20

0.05

Figure 4-8 Continuous uniform

probability density function.

a

1

b – a

x

f(x)

b

4-31. Suppose Xhas a continuous uniform distribution over

the interval [1.5, 5.5].

(a) Determine the mean, variance, and standard deviation of X.

(b) What is?

4-32. Suppose Xhas a continuous uniform distribution over

the interval 3 1, 1 4.

P 1 X2.5 2

(a) Determine the mean, variance, and standard deviation of X.

(b) Determine the value for xsuch that P(xXx) 0.90.

4-33. The net weight in pounds of a packaged chemical her-

bicide is uniform for pounds.

(a) Determine the mean and variance of the weight of pack-

ages.

49.75x50.25

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