60 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Example 2.33
e PDF of the diameter in millimeter of a shaft is:
f .x/ D
8
<
:
0 x < 25:4 mm
18e
18.x25:4/
x 25:4 mm:
Calculate the mean of the shaft and its cross-section area.
Solution:
Per Equation (2.44), the mean of the shaft is:
x
D E
.
x
/
D
Z
1
1
xf
.
x
/
dx D
Z
25:4
0
x 0dx C
Z
1
25:4
x18e
18
.
x25:4
/
dx
D 0 C
xe
18
.
x25:4
/
1
18
e
18
.
x25:4
/
ˇ
ˇ
ˇ
ˇ
1
25:4
D 25:4 C
1
18
D 25:46 .mm/:
e cross-section area of the shaft A
.
x
/
is a function of the diameter, which is a random variable
x, has the following function:
A
.
x
/
D
4
x
2
:
Per Equation (2.46), the mean of the cross-section area of the shaft A
.
x
/
is
A
.
x
/
D E
Œ
A
.
x
/
D
Z
1
1
A
.
x
/
f
.
x
/
dx
D
Z
25:4
1
4
x
2
0dx C
Z
1
25:4
4
x
2
18e
18
.
x25:4
/
dx
D 0 C
4
x
2
e
18
.
x25:4
/
2
18
xe
18
.
x25:4
/
2
18 18
e
18
.
x25:4
/
ˇ
ˇ
ˇ
ˇ
1
25:4
D 508:93
mm
2
:
2.11 STANDARD DEVIATION AND COEFFICIENT OF
VARIANCE
Before we can provide the formula for the standard deviation of a random variable, we need to
define the variance of it first.
e variance of a random variable X is the mean of the function
.
X
x
/
2
and is defined as
Var
.
X
/
D E
h
.
X
x
/
2
i
D E
X
2
2X
x
C
2
x
D E
X
2
2
x
E
Œ
X
C E
2
x
D E
X
2
2
2
x
C
2
x
D E
X
2
2
x
; (2.48)
2.11. STANDARD DEVIATION AND COEFFICIENT OF VARIANCE 61
where Var
.
X
/
refers to the variance of a random variable X and
x
is the mean of a random
variable X.
For a continuous random variable, the variance of a random variable X is
Var
.
X
/
D E
X
2
2
x
D
Z
1
1
x
2
f
.
x
/
dx
2
x
: (2.49)
For a discrete random variable, the variance of a random variable X is
Var
.
X
/
D E
X
2
2
x
D
X
All i
x
2
i
p
.
x
i
/
2
x
: (2.50)
In the previous section, Equation (2.28) is used to calculate the standard deviation of sampling
data. After the definition of the PDF and the PMF are defined, the variance of a random variable
can define the standard deviation.
Standard deviation is a measure of variation or dispersion of a set of data values around its
central value and is defined as the square root of the variance of a random variable.
For a continuous random variable, the standard deviation of a random variable X is
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
s
Z
1
1
x
2
f
.
x
/
dx
2
x
: (2.51)
For a discrete random variable, the standard deviation of a random variable X is
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
s
X
All i
x
2
i
p
.
x
i
/
2
x
; (2.52)
where
x
and Var
.
X
/
are the standard deviation and the variance of the random variable X,
respectively. f
.
x
/
is the PDF of a continuous random variable X. p
.
x
i
/
is the PMF of a discrete
random variable X .
x
is the mean of random variable X.
After the mean and standard deviation of a random variable X have been determined, we
can define the coefficient of variance.
e coefficient of variance is the ratio of the standard deviation to the mean of a random variable
and can be directly calculated by Equation (2.29), from page 44, which is repeated here:
x
D
x
x
; (2.29)
where
x
is the coefficient of variance of a random variable x.
x
and
x
are the standard devi-
ation and the mean of a random variable X.
62 2. FUNDAMENTAL RELIABILITY MATHEMATICS
Example 2.34
e PDF of the diameter in millimeter of a shaft is:
f
.
x
/
D
8
<
:
0 x < 20 mm
15e
15.x20/
x 20 mm:
Calculate its mean, standard deviation, and coefficient of variance.
Solution:
Per Equation (2.43), the mean of the shaft’s diameter X is
x
D E
.
X
/
D
Z
1
1
xf
.
x
/
dx D
Z
20
0
x 0dx C
Z
1
20
15xe
15
.
x20
/
dx
D 0 C
xe
15
.
x20
/
1
15
e
15
.
x20
/
ˇ
ˇ
ˇ
ˇ
1
20
D 20 C
1
15
D 20:667 .mm/
E
X
2
D
Z
1
1
x
2
f
.
x
/
dx D
Z
20
0
x
2
0dx C
Z
1
20
15x
2
e
15
.
x20
/
dx
D 0 C
x
2
e
15
.
x20
/
2
15
xe
15
.
x20
/
2
15 15
e
15
.
x20
/
ˇ
ˇ
ˇ
ˇ
1
20
D 20
2
C
2
15
20 C
2
15 15
D 402:676 .mm
2
/:
Per Equation (2.50), the standard deviation of the shaft’s diameter is
x
D
p
Var
.
X
/
D
q
E
Œ
X
2
2
x
D
p
402:676 20:667
2
D 0:0667 .mm/:
Per Equation (2.28), the coefficient of variance is:
x
D
x
x
D
0:0667
20:0667
D 0:00332:
Example 2.35
e number of automobiles arriving at a tollbooth per minute has the distribution in Table 2.8.
Determine the mean, standard deviation, and coefficient of variance.
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