11.2 Basic Concepts of Probability Theory 635
possible value isT, then
FX(x) = 0 for allx < S and FX(x) = 1 for allx>T
Probability Density Function (Continuous Case). The probability density function
of a random variable is defined by
fX(x) dx =P(x≤X≤x+dx) (11.6)
which is equal to the probability of detectingXin the infinitesimal interval (x, x+
dx). The distribution function ofXis defined as the probability of detectingXless
than or equal tox, that is,
FX(x)=
∫x
−∞
fX(x′) dx′ (11.7)
where the conditionFX(– ∞)= 0 has been used. As the upper limit of the integral
goes to infinity, we have ∫
∞
−∞
fX(x) dx=FX(∞)= 1 (11.8)
This is called thenormalization condition. A typical probability density function and
the corresponding distribution functions are shown in Fig. 11.1.
11.2.3 Mean and Standard Deviation
The probability density or distribution function of a random variable contains all the
information about the variable. However, in many cases we require only the gross
properties, not entire information about the random variable. In such cases one computes
only the mean and the variation about the mean of the random variable as the salient
features of the variable.
Mean. Themean value(also termed the expected value oraverage) is used to
describe the central tendency of a random variable.
Figure 11.1 Probability density and distribution functions of a continuous random variableX:
(a)density function; (b)distribution function.
