Applied Statistics and Probability for Engineers

100 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

the same function, f(x), over the smaller interval. By appropriate choice of the shape of f(x),

we can represent the probabilities associated with any continuous random variable X. The

shape of f(x) determines how the probability that Xassumes a value in [14.5 mA, 14.6 mA]

compares to the probability of any other interval of equal or different length.

For the density function of a loading on a long thin beam, because every point has zero

width, the loading at any point is zero. Similarly, for a continuous random variable Xand any

value x.

Based on this result, it might appear that our model of a continuous random variable is use-

less. However, in practice, when a particular current measurement is observed, such as 14.47

milliamperes, this result can be interpreted as the rounded value of a current measurement that

is actually in a range such as Therefore, the probability that the

rounded value 14.47 is observed as the value for Xis the probability that Xassumes a value in

the interval [14.465, 14.475], which is not zero. Similarly, because each point has zero

probability, one need not distinguish between inequalities such as or for continuous

random variables.

14.465x14.475.

P 1 Xx 2  0

If Xis a continuous random variable,for any and

P 1 x 1 Xx 22 P 1 x 1 Xx 22 P 1 x 1 Xx 22 P 1 x 1 Xx 22 (4-2)

x 1 x 2 ,

EXAMPLE 4-1 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 for What is the probability that a current meas-

urement is less than 10 milliamperes?

The probability density function is shown in Fig. 4-4. It is assumed that wherever

it is not specifically defined. The probability requested is indicated by the shaded area in Fig. 4-4.

P 1 X 102 

10

0

f 1 x 2 dx

10

0

0.05 dx0.5

f 1 x 2  0

f 1 x 2 0.05 0 x20.

Figure 4-4 Probability density

function for Example 4-1.

01020 x

0.05

f (x)

Figure 4-3 Histogram approximates a probability density

function.

x

f (x)

c 04 .qxd 5/10/02 5:19 PM Page 100 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: