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4-2 PROBABILITY DISTRIBUTIONS AND PROBABILITY DENSITY FUNCTIONS 99

f (x)
Loading P(a < X < b)

x a b x
Figure 4-1 Density Figure 4-2 Probability determined from the area
function of a loading on a under f(x).
long, thin beam.

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 variable X. 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 X is be-
tween a and b is determined as the integral of f(x) from a to b. See Fig. 4-2.

Definition
For a continuous random variable X, a probability density function is a function
such that
(1) f 1x2 0

(2) f 1x2 dx 1

b

(3) P1a X b2 f 1x2 dx area under f 1x2 from a to b

a
for any a and b (4-1)

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 f 1x2 dx 1,

0 P1a X b2 1 so that the probabilities are properly restricted. A probability density
function is zero for x values that cannot occur and it is assumed to be zero wherever it is not
specifically defined.
A histogram is 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 area that represents the prob-
ability that X assumes a value in [a, b]. For the current measurement example, the proba-
bility that X results in [14 mA, 15 mA] is the integral of the probability density function of
X over this interval. The probability that X results in [14.5 mA, 14.6 mA] is the integral of