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
Z
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 conFdence intervals on the
ranges of standard normal variates. ±or example, we can assert that we are 95
percent conFdent that
z
will fall in the interval [
−
1
.
96
,
1
.
96]. There are inFnitely
many 95% conFdence intervals that we could provide, but this one is the smallest.
Only the standard Normal distribution is tabulated, because a non-standard
Normal variate can be standardised; and hence the conFdence 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 deFned in terms of
x
, there
is an equivalent event
B
deFned 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
Fnd the the p.d.f of
y
denoted by
g
(
y
).
±irst, consider the discrete random variable
x
∼
f
(
x
). Then, if
y
∼
g
(
y
), it
must be the case that
X
y
∈
B
g
(
y
X
x
∈
A
f
(
x
)
.
But we may express
x
as a function of
y
, denoted by
x
=
x
(
y
). Therefore,
X
y
∈
B
g
(
y
X
y
∈
B
f
{
x
(
y
)
}
,
whence
g
(
y
f
{
x
(
y
)
}
.