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108 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

f(x)
f(x)
1
b – a
0.05

a b x 0 5 10 15 20 x
Figure 4-8 Continuous uniform Figure 4-9 Probability for Example 4-9.
probability density function.

EXAMPLE 4-9 Let the continuous random variable X denote the current measured in a thin copper wire in
milliamperes. Assume that the range of X is [0, 20 mA], and assume that the probability den-
sity function of X is f 1x2 0.05, 0 x 20.
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.

10

P15 X 102 f 1x2 dx

5
510.052 0.25

The mean and variance formulas can be applied with a 0 and b 20. Therefore,

2
E1X2 10 mA and V1X2 20 12 33.33 mA 2
Consequently, the standard deviation of X is 5.77 mA.

The cumulative distribution function of a continuous uniform random variable is ob-
tained by integration. If a x b,

x

F1x2 1 1b a2 du x 1b a2 a 1b a2
a
Therefore, the complete description of the cumulative distribution function of a continuous
uniform random variable is

0 x a
F1x2 • 1x a2 1b a2 a x b
1 b x

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

EXERCISES FOR SECTION 4-5
4-31. Suppose X has a continuous uniform distribution over (a) Determine the mean, variance, and standard deviation of X.
the interval [1.5, 5.5]. (b) Determine the value for x such that P( x X x) 0.90.
(a) Determine the mean, variance, and standard deviation of X. 4-33. The net weight in pounds of a packaged chemical her-
(b) What is P1X 2.52 ? bicide is uniform for 49.75 x 50.25 pounds.
4-32. Suppose X has a continuous uniform distribution over (a) Determine the mean and variance of the weight of pack-
the interval 3 1, 14. ages.