PROBABILITY DISTRIBUTIONS: DISCRETE AND CONTINUOUS
Univariate Probability Distributions.
Let
S
be a sample space with a prob-
ability measure
P
defned over it, and let
x
be a real scalar-valued set Function
defned over
S
, which is described as a random variable.
IF
x
assumes only a fnite number oF values in the interval [
a, b
], then it is said
to be discrete in that interval. IF
x
assumes an infnity oF values between any two
points in the interval, then it is said to be everywhere continuous in that interval.
Typically, we assume that
x
is discrete or continuous over its entire domain, and
we describe it as discrete or continuous without qualifcation.
IF
x
∈{
x
1
,x
2
,...
}
, is discrete, then a Function
f
(
x
i
) giving the probability
that
x
=
x
i
is called a
probability mass function.
Such a Function must have the
properties that
f
(
x
i
)
≥
0
,
For all
i,
and
X
i
f
(
x
i
)=1
.
Example.
Consider
x
0
,
1
,
2
,
3
}
with
f
(
x
)=(1
/
2)
x
+1
. It is certainly true
that
f
(
x
i
)
≥
0 For all
i
. Also,
X
i
f
(
x
i
)=
½
1
2
+
1
4
+
1
8
+
1
16
+
···
¾
=1
.
To see this, we may recall that
1
1
−
θ
=
{
1+
θ
+
θ
2
+
θ
3
+
···}
,
whence
θ
1
−
θ
=
{
θ
+
θ
2
+
θ
3
++
θ
4
+
.
Setting
θ
/
2 in the expression above gives
θ/
(1
−
θ
1
2
/
(1
−
1
2
,
which is
the result that we are seeking.
IF
x
is continuous, then a
probability density function
(p.d.F.)
f
(
x
) may be defned
such that the probability oF the event
a<x
≤
b
is given by
P
(
≤
b
Z
b
a
f
(
x
)
dx.
Notice that, when
b
=
a
, there is
P
(
x
=
a
Z
a
a
f
(
x
)
dx
=0
.
That is to say, the integral oF the continuous Function
f
(
x
) at a point is zero. This
corresponds to the notion that the probability that the continuous random variable
1