PROBABILITY DISTRIBUTIONS: (continued)
The Standard Normal Distribution.
Consider the function
g
(
z
) =
e
−
z
2
/
2
,
where
−∞
< z <
∞
. There is
g
(
z
)
>
0 for all
z
and also
e
−
z
2
/
2
dz
=
1
√
2
π
.
It follows that
f
(
z
) =
1
√
2
π
e
−
z
2
/
2
constitutes a p.d.f., which we call the standard normal and which we may denote
by
N
(
z
;0
,
1). The normal distribution, in general, is denoted by
N
(
x
;
µ, σ
2
); so, in
this case, there are
µ
= 0 and
σ
2
= 1.
The standard normal distribution is tabulated in the back of virtually every
statistics textbook. Using the tables, we may establish confidence intervals on the
ranges of standard normal variates. For example, we can assert that we are 95
percent confident that
z
will fall in the interval [
−
1
.
96
,
1
.
96]. There are infinitely
many 95% confidence intervals that we could provide, but this one is the smallest.
Only the standard Normal distribution is tabulated, because a nonstandard
Normal variate can be standardised; and hence the confidence interval for all such
variates can be obtained from the
N
(0
,
1) tables.
The change of variables technique.
Let
x
be a random variable with a known
p.d.f.
f
(
x
) and let
y
=
y
(
x
) be a monotonic transformation of
x
such that the
inverse function
x
=
x
(
y
) exists. Then, if
A
is an event defined in terms of
x
, there
is an equivalent event
B
defined in terms of
y
such that if
x
∈
A
, then
y
=
y
(
x
)
∈
B
and
vice versa.
Then,
P
(
A
) =
P
(
B
) and, under very general conditions, we can
find the the p.d.f of
y
denoted by
g
(
y
).
First, consider the discrete random variable
x
∼
f
(
x
). Then, if
y
∼
g
(
y
), it
must be the case that
y
∈
B
g
(
y
) =
x
∈
A
f
(
x
)
.
But we may express
x
as a function of
y
, denoted by
x
=
x
(
y
). Therefore,
y
∈
B
g
(
y
) =
y
∈
B
f
{
x
(
y
)
}
,
whence
g
(
y
) =
f
{
x
(
y
)
}
.
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 Spring '12
 D.S.G.Pollock
 Variance, Probability theory, probability density function, variates

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