76 2. FUNDAMENTAL RELIABILITY MATHEMATICS
In MATLAB, the commands for calculating the PDF and CDF of a standard normally dis-
tributed variable are:
.
x
/
D normpdf .z/ (2.92)
ˆ
.
z
/
D P
.
Z z
/
D normcdf .z/: (2.93)
Example 2.41
A random variable Z can be assumed to follow the standard normal distribution. Use Excel and
MATLAB to calculate the probability of P
.
3 Z 3
/
.
Solution:
By a Microsoft Excel function, that is, per Equation (2.91), we have:
P
.
3 Z 3
/
D ˆ
.
3
/
ˆ
.
3
/
D NORM:S:DIST
.
3; true
/
NORM:S:DIST
.
3; true
/
D 0:9987 0:0013 D 0:9973:
By the MATLAB program, that is, per Equation (2.93), we have:
P
.
3
Z
3
/
D
ˆ
.
3
/
ˆ
.
3
/
D
normcdf
.
3
/
normcdf
.
3
/
D 0:9987 0:0013 D 0:9973:
Since the PDF of a standard normal distribution is symmetrical about the vertical axis,
that is, z D D 0, as shown in Figure 2.22, the shaded area depicted in Figure 2.22 will be the
same. e left shaded area is the probability P
.
Z z
/
, that is,
P
.
Z z
/
D ˆ
.
z
/
: (2.94)
e right shaded area is the probability P
.
Z > z
/
, that is,
P
.
Z > z
/
D 1 P
.
Z z
/
D 1 ˆ
.
z
/
: (2.95)
Since both shaded areas are the same as shown in Figure 2.22, we have the following equation
per Equations (2.94) and (2.95):
ˆ
.
z
/
D 1 ˆ
.
z
/
: (2.96)
2.12. SOME TYPICAL PROBABILITY DISTRIBUTIONS 77
0-3-4 -2 -1
-z z
1 2 3 4
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
z -e Standard Normal Variable
f (z)-Probability Density
Function
Figure 2.22: Schematic of P .Z z/ and P .Z z/.
Standard normal distribution table: e CDF of a standard normal variable Z, named as the
standard normal distribution table, is shown in Table 2.10. is table can be used to determine
the CDF ˆ
.
z
/
of the standard normal variable for the value of 0 z 3. For the value of
3 z 0, per Equation (2.96), we can also determine the CDF ˆ
.
z
/
of a standard normal
variable.
For a general normally distributed variable, we can use the following equation to convert
it into the standard normal distributed variable. For X D N.
x
;
x
/, Z will be the standard
normal distributed variable if Z is defined by:
Z D
X
x
x
: (2.97)
Per Equation (2.97), we can have the following two equations to calculate the CDF or the prob-
ability of a normally distributed variable by using the standard normal distribution Table 2.10:
F
.
x
/
D P
.
X x
/
D ˆ
x
x
x
(2.98)
P
.
a X b
/
D ˆ
b
x
x
ˆ
a
x
x
: (2.99)
Example 2.42
If the tensile yield strength of a ductile material follows a normal distribution with a mean D
61:2 (ksi) and standard deviation D 4:25 (ksi). Use the standard normal distribution Table 2.10
to calculate probability P .50 X 70/.
78 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Table 2.10: e standard normal distribution table
0-3 -2 -1 1 2 3
Φ(z) = P (Z ≤ z)
=
1
1
2
x
–∞
π 2
Standard Normal Distribution
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