16)
It is confined in the interval , with the additional constrains of and
.
The derivative of the CDF is described by the probability density function (PDF), which is defined as
(1.17)
The PDF is always a positive function with a total area equal to 1:
(1.18)
Now, using the earlier definitions, we have rewritten the relationship between the CDF and the PDF.
(1.19)
Next we present a few methods used to analyze a random variable. The mean or expected value
of a random variable X is defined as
(1.20)
E{g} is the expected value of the random variable g. The expected value is a linear operator with the
following properties (assuming K is a constant):
(1.21)
(1.22)
The variance of a random variable is defined as
(1.23)
which is also represented in the following manner (given the above definitions).
(1.24)
Assuming the mean value is a constant, we then obtain
(1.25)
Var{g} is the variance of the random variable g. The variance is used to measure the concentration
(or variability) of g around its expected value. The square root of the variance is called the standard
deviation of the random variable being analyzed.
Acommonly used random variable is the Gaussian or normal random variable. The PDF is defined as
(1.
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