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/.