Random Variable - Random Signals and Processes Primer with MATLAB - page 35

2.2 Distribution Function

2.2.1 Definition

In order to completely describe a random variable, it is necessary to know not only
all of the possible values it takes but also how often it takes these values. In other
words, it is necessary to know the corresponding probabilities.
The probability that random variable X has values less than or equal to x is called
distribution function or distribution ,

F X ðxÞ¼PfX xg; 1< x <1 (2.10)

Some authors also use the name cumulative distribution function ( CDF ).
This term includes both the discrete and continuous variables.
In the following examples, the calculation of the distribution functions is
demonstrated. As defined in ( 2.10 ), the distribution is defined for all values of x .
First, we divide all the x -axis into corresponding subintervals as defined by the
corresponding values of the random variable. Next, we find the corresponding
probability for each subinterval.
The following example illustrates the calculation of the distribution function for
a discrete random variable.

Example 2.2.1 Consider a discrete random variable with only two values, 0 and 1
(Fig. 2.8a ). The corresponding probabilities are:

PfX ¼ 1 g¼p ¼ 3 = 4 ;

PfX ¼ 0 g¼q ¼ 1 = 4 : (2.11)

Fig. 2.8 Illustration of the distribution in Example 2.2.1