Digital Signal Processing Reference
In-Depth Information
A.2.6. NO. A random variable
X
represents the function or rule which assigns a
real number to each outcome
s
i
in a sample space
S.
However, there is
nothing random in
X
(
s
i
) because this function is fixed and deterministic,
and is usually chosen by us [LEO94, p. 85], [KAY06, p. 107].
What is truly random is the input argument
s
i
and consequently the
random output of the function. However, traditionally, this terminology is
widely accepted.
A.2.7. NO
.
In order to describe the r.v. completely, we must also know how often
the values of
x
are taken. That is we must include probabilistic measure.
A.2.8. Every random variable uniquely defines its distribution. Each random
variable can have only one distribution. However, there is an arbitrary
number of different random variables which have the same distribution.
Let, for example,
X
be a discrete r.v. which takes only two values:
1 and 1,
each with a probability of 1/2. Consider the variable
Y ¼X
. It is obvious
that
X
and
Y
are different variables. However, both of these variables have
the same distribution [GNE82, p. 130]:
8
<
0
for
x <
1
;
F
X
ðxÞ¼
1
=
2
for
1
x
1
;
:
1
for
x >
1
:
8
<
(2.566)
0
for
y <
1
;
F
Y
ðyÞ¼
1
=
2
for
1
y
1
;
:
1
for
y >
1
:
A.2.9. The definition of the distribution function
F
X
(
x
)
¼ P
{
X x
} includes
both cases: the discrete and continuous random variables. However, for
continuous random variables, the probability
P
{
X ¼ x
} is zero, and we
may write
F
X
(
x
)
¼ P
{
X < x
}.
A.2.10. The PDF of the discrete random variables has only delta functions which
are, by definition, continuous functions. Therefore, the PDF of the discrete
random variable is a continuous function.
A.2.11. The distribution function of a discrete random variable
X
is a discontinuous
step function, with jumps at the discrete values of
x
which are equal to the
corresponding probabilities. However, between the jumps, the function
F
X
(
x
) remains constant, and the distribution function is “continuous from
the right.”
For continuous random variables, the distribution function is continuous
for any
x
and in addition, it has a derivative everywhere except possibly at
certain points.
If a random variable is mixed, then its distribution function is continu-
ous in certain intervals and has discontinuities (jumps) at particular points
[WEN86, p. 112].
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