The Gaussian or Normal Probability Density Function
Author: John M. Cimbala, Penn State University
Latest revision: 07 September 2007
The Gaussian or Normal Probability Density Function
•
Gaussian or normal PDF
– The
Gaussian probability density function
(also called the
normal probability
density function
or simply the
normal PDF
) is
the vertically normalized PDF that is produced from a
signal or measurement that has purely random errors
.
o
The normal probability density function is
()
2
2
2
2
2
11
ee
x
p
2
22
x
x
fx
μ
σ
σπ
−−
⎛⎞
⎜⎟
==
⎝⎠
.
o
Here are some of the properties of this special distribution:
±
It is symmetric about the mean.
f
(
x
)
x
±
The mean, median, and mode are all equal
to
, the
expected value
(at the peak of the
distribution).
Small
±
Its plot is commonly called a “
bell curve
”
because of its shape.
±
The actual shape depends on the magnitude
of the standard deviation. Namely, if
is
small, the bell will be tall and skinny, while
if
is large, the bell will be short and fat, as
sketched.
•
Standard normal density function
– All of the Gaussian PDF cases, for
any
mean value and for
any
standard
deviation, can be collapsed into
one normalized curve
called the
standard normal density function
.
o
This normalization is accomplished through the variable transformations introduced previously, i.e.,
x
z
−
=
and
() ()
f
zf
=
x
, which yields
2
/2
2
exp
/ 2
z
fz
e
z
ππ
−
=
−
.
This standard normal density function is valid for
any
signal measurement, with
any
mean, and with
any
standard deviation, provided that the errors
(deviations) are
purely random
.
o
A plot of the standard normal density function w
generated in Excel, using the above equation for
f
(
z
). It is shown to the right.
as
o
It turns out that
the probability that variable
x
lies
between some range
x
1
and
x
2
is the
same
as the
probability that the transformed variable
z
lies
between the corresponding range
z
1
and
z
2
, where
z
is the transformed variable defined above. In other words,
( ) ( )
121
Px x x
Pz z z
<≤ =
<≤
2
where
1
1
x
z
−
=
and
2
2
x
z
−
=
.
o
Note that
z
is
dimensionless
, so there are no units to worry about, so long as the mean and the standard
deviation are expressed in the
same units
.
o
Furthermore, since
(
2
1
12
x
x
f xd
x
∫
)
, it follows that
(
2
1
z
z
)
f zd
z
∫
.
o
We define
A
(
z
)
as
the area under the curve between 0 and z
, i.e., the special case where
z
1
= 0 in the
above integral, and
z
2
is simply
z
. In other words,
A
(
z
) is
the probability that a measurement lies
between 0 and z
, or
0
z
A
z
=
∫
d
z
, as illustrated on the graph below.
Large