Applied Statistics and Probability for Engineers

4-2 PROBABILITY DISTRIBUTIONS AND PROBABILITY DENSITY FUNCTIONS 99

under the density function over this interval, and it can be loosely interpreted as the sum of all

the loadings over this interval.

Similarly, a probability density function f(x) can be used to describe the probability dis-

tribution of a continuous random variableX. If an interval is likely to contain a value for X,

its probability is large and it corresponds to large values for f(x). The probability that Xis be-

tween aand bis determined as the integral of f(x) from ato b. See Fig. 4-2.

For a continuous random variable X, a probability density functionis a function

such that

(1)

(2)

(3) area under from ato b

for any aand b (4-1)

P 1 aXb 2  f 1 x 2

b

a

f 1 x 2 dx







f 1 x 2 dx 1

f 1 x 2  0

Definition

Loading

x

P(a < X < b)

ab x

f (x)

Figure 4-1 Density

function of a loading on a

long, thin beam.

Figure 4-2 Probability determined from the area

under f(x).

A probability density function provides a simple description of the probabilities associ-

ated with a random variable. As long as f(x) is nonnegative and

so that the probabilities are properly restricted. A probability density

function is zero for xvalues that cannot occur and it is assumed to be zero wherever it is not

specifically defined.

A histogramis an approximation to a probability density function. See Fig. 4-3. For each

interval of the histogram, the area of the bar equals the relative frequency (proportion) of the

measurements in the interval. The relative frequency is an estimate of the probability that a

measurement falls in the interval. Similarly, the area under f(x) over any interval equals the

true probability that a measurement falls in the interval.

The important point is that f(x) is used to calculate an areathat represents the prob-

ability that Xassumes a value in [a,b]. For the current measurement example, the proba-

bility that Xresults in [14 mA, 15 mA] is the integral of the probability density function of

X over this interval. The probability that Xresults in [14.5 mA, 14.6 mA] is the integral of

0 P 1 aXb 2  1

 f^1 x^2 dx1,

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