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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