Digital Signal Processing Reference
In-Depth Information
A.2.12. By definition, the distribution function exists for any random variable,
whether discrete, continuous, or mixed, for all values of
x
.
A.2.13. For discrete variable, the probability that the variable takes a particular
x
i
is
the finite number
P
(
x
i
). However, in the PDF, the probability is defined as
an area below the PDF in a given interval. Then, the interval is approaching
zero for the specific point
x
i
. In order for the probability (the area) to
remain the same, the height must approach infinity.
A.2.14. Zero. The probability that the continuous random variable takes a particu-
lar value in its range is zero because the continuous range is enumerable.
However, there is a nonzero probability that the random variable takes
a value in the infinitesimal interval d
x
around the particular value
x
,
P
{
x < X x
+d
x
}
¼ f
X
(
x
)d
x
, where
f
X
(
x
) is the value of the PDF of
the r.v.
X
for
x.
However, as
f
X
(
x
) is a finite value for a continual r.v. and
d
x ¼
0 in the discrete point,
P
{
X ¼ x
} must be also zero.
A.2.15. Variance does not exist for all random variables as, for example, in the case
of a Cauchy random variable. For a more detailed discussion, see [KAY06,
p. 357].
A.2.16. This is true for a PDF which has only one peak (an unimodal PDF).
However, if a PDF has two peaks (a bimodal PDF), the standard deviation
measures the range of the most expected values and not the widths of any
of the peaks [HEL91, pp. 113-114].
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