Databases Reference
In-Depth Information
ExampleA.3.1:
Assuming that the frequency of occurrence was an accurate estimate of the probabilities, let
us obtain the
cdf
for our television example:
⎧
⎨
<
0
x
0
.
<
0
40
x
1
0
.
81
x
<
2
0
.
802 2
x
<
3
0
.
822 3
x
<
4
F
X
(
x
)
=
0
.
912 4
x
<
5
⎩
0
.
932 5
x
<
6
0
.
972 6
x
<
7
0
.
99
7
x
<
8
1
.
00
8
x
Notice a few things about this
cdf
. First, the
cdf
consists of step functions. This is
characteristic of discrete random variables. Second, the function is continuous from the right.
This is due to the way the
cdf
is defined.
The
cdf
is somewhat different when the random variable is a continuous random variable.
For example, if we sample a speech signal and then take the differences of the samples, the
resulting random process would have a
cdf
that looks something like this:
1
2
e
2
x
x
0
F
X
(
x
)
=
1
2
e
−
2
x
1
−
x
>
0
The thing to notice in this case is that because
F
X
(
x
)
is continuous
x
−
)
=
P
(
X
=
x
)
=
F
X
(
x
)
−
F
X
(
0
We can also have processes that have distributions that are continuous over some ranges and
discrete over others.
Along with the cumulative distribution function, another function that also comes in very
handy is the
probability density function (pdf)
.The
pdf
corresponding to the
cdf F
X
(
x
)
iswritten
as
f
X
(
. For continuous
cdf
s, the
pdf
is simply the derivative of the
cdf
. For discrete random
variables, taking the derivative of the
cdf
introduces delta functions, which have problems of
their own. So in the discrete case, we obtain the
pdf
through differencing. It is somewhat
awkward to have different procedures for obtaining the same function for different types of
random variables. It is possible to define a rigorous unified procedure for getting the
pdf
from
the
cdf
for all kinds of random variables. However, in order to do so, we need some familiarity
with measure theory, which is beyond the scope of this appendix. Let us look at some examples
of
pdf
s.
x
)
