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 scalarvalued 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
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assumes a particular value amongst the non denumerable (i.e. uncountable) in
fnity oF values within its domain must be zero. Nevertheless, the value oF
f
(
x
)at
a point continues to be oF interest; and we describe this as a probability measure
as opposed to a probability.
Notice also that it makes no di±erence whether we include oF exclude the
endpoints oF the interval [
a, b
]=
{
a
≤
x
≤
b
}
. The reason that we have chosen to
use the halFopen halFclosed interval (
a, b
{
a<x
≤
b
}
is that this accords best
with the defnition oF a defnite integral over [
a, b
], which is obtained by subtracting
the integral over (
−∞
,a
] From the integral over (
−∞
,b
].
Example.
Consider the exponential Function
f
(
x
)=
1
a
e
−
x/a
defned over the interval [0
,
∞
) and with
α>
0. There is
f
(
x
)
>
0 and
Z
∞
0
1
a
e
−
x/a
dx
=
h
−
e
−
x/a
i
∞
0
=1
.
ThereFore, this Function constitutes a valid p.d.F. The exponential distribution
provides a model For the liFespan oF an electronic component, such as Fuse, For
which the probability oF Failing in the ensuing period is liable to be independent oF
how long it has survived already.
Corresponding to any p.d.F
f
(
x
), there is a cumulative distribution Function,
denoted by
F
(
x
), which, For any value
x
∗
, gives the probability oF the event
x
≤
x
∗
.
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 Spring '12
 D.S.G.Pollock
 Probability theory, Probability mass function, Fill

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