72 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Normal distribution: e PDF of a normally distributed random variable X , also known as
a normal distribution or Gaussian distribution is:
f
X
.
x
/
D
1
p
2
x
exp
"
1
2
x
x
x
2
#
1 < x < 1; (2.74)
where
x
and
x
are the mean and standard deviation of a normally distributed random variable,
respectively.
e CDF of a normally distributed random variable X is
F
X
.
x
/
D P
.
X x
/
D
Z
x
1
1
p
2
x
exp
"
1
2
x
x
x
2
#
dx 1 < x < 1: (2.75)
If X follows a normal distribution with distribution parameters
x
and
x
, we have the following
equations:
E
.
X
/
D
x
(2.76)
x
D E
h
.
X
x
/
2
i
D
p
Var.X/D
x
: (2.77)
When two distribution parameters of a normal distribution
x
and
x
are known, the PDF of
this normal distribution is fully specified, as shown in Equation (2.74). A normal distribution
of a random variable with distribution parameters
x
and
x
can be expressed as
X D N
.
x
;
x
/
: (2.78)
e PDFs of a normal distribution with different distribution parameters are plotted in Fig-
ure 2.20. From Figure 2.20, the PDF of normal distribution has a bell-shaped curve with the
symmetrical line x D
x
. e mean
x
of normal distribution will control the horizontal loca-
tion of the bell-shaped curve and the standard deviation
x
will control the shape. When the
means are the same, the bell-shaped curve will be thinner with a smaller standard deviation.
e CDF in Equation (2.75) of a normal distribution does not have an explicit theoretical
solution. However, it can be easily calculated by using Excel and MATLAB.
In Microsoft Excel, the functions for calculating the PDF and the CDF of a normal
distribution are:
f
.
x
/
D NORM:DIST
.
x;
x
;
x
; false
/
(2.79)
F
.
x
/
D P
.
X x
/
D NORM:DIST
.
x;
x
;
x
; true
/
: (2.80)
In MATLAB, the commands for calculating the PDF and CDF of a normal distribution are:
f
.
x
/
D normpdf
.
x;
x
;
x
/
(2.81)
F
.
x
/
D P
.
X x
/
D normcdf
.
x;
x
;
x
/
: (2.82)