2.10. MEAN OF A RANDOM VARIABLE 57
where
P
i
p.x
i
/ is the sum of probabilities at the point where x
i
is less than or equal to the value
of x.
Example 2.30
Determine and plot the PMF and the CDF of a discrete random variable of rolling a dice.
Solution:
For rolling a dice, the discrete random variable X will have possible value x D 1; 2; 3; 4; 5, and
6. Since each side of a die is equally likely to show up, the probability of realizing any number
between 1 and 6 is 1/6. So, the PMF with be:
p
.
1
/
D p
.
2
/
D p
.
3
/
D p
.
4
/
D p
.
5
/
D p
.
6
/
D
1
6
:
Per Equation (2.42), the CDF of rolling a dice will be:
F
.
x
/
D P
.
X < x
/
D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
0 x < 1
1=6 1 x < 2
2=6 2 x < 3
3=6 3 x < 4
4=6 4 x < 5
5=6 5 x < 6
6=6 6 x:
e plots of the PMF and the CDF, in this case, are shown in Figures 2.14 and 2.15,
respectively. It can be seen that the PMF is discrete dots because the PMF has only values at
discrete points. At all other points, the PMF is zero. However, the CDF is step-curve. For
example, the PMF is zero at x D 4:5, but the PDF at x D 4:5 is 4/6.
2.10 MEAN OF A RANDOM VARIABLE
e PDF function for a continuous random variable and the PMF for a discrete random variable
contain all the information about this random variable. erefore, they can be used to determine
the mean and standard deviation. en, the coefficient of the variance can be calculated.
Mean, termed as the expected value, mathematical expectation, or average, has been defined in
Section 2.7. It is a measure of the central value of a random variable and can be determined by
the following equations.
For a continuous random variable X, the mean
x
alternatively, E.x/ is
x
D E
.
x
/
D
Z
1
1
xf .x/dx; (2.44)
58 2. FUNDAMENTAL RELIABILITY MATHEMATICS
1/6
0
x-Discrete Random Variable of Rolling a Dice
p (x)-Probability Mass
Function
0 1 2 3 4 5 6 7
Figure 2.14: e plot of the PMF of rolling a dice.
1
5/6
2/3
1/2
1/3
1/6
0
x-Discrete Random Variable of Rolling a Dice
p (x)-Probability Mass
Function
0 2 4 6
Figure 2.15: e plot of the CDF of rolling a dice.
where
f .x/
is the PDF of a continuous random variable
x
.
For a discrete random variable X , the mean
x
alternatively, E.x/ is
x
D E
.
x
/
D
X
All i
x
i
p
.
x
i
/
; (2.45)
where p
.
x
i
/
is the PMF of a discrete random variable at the point x
i
.
According to the definition of a random variable, any function h
.
x
/
of a random variable
x will be itself a random variable. e mean of h
.
x
/
is defined as:
For a continuous random variable,
h.x/
D E
Œ
h.x/
D
Z
1
1
h.x/f .x/dx: (2.46)
For a discrete random variable,
h.x/
D E
Œ
h.x/
D
X
All i
h
.
x
i
/
p
.
x
i
/
: (2.47)
2.10. MEAN OF A RANDOM VARIABLE 59
Example 2.31
Calculate the mean of a discrete random variable of rolling dice.
Solution:
For this discrete random variable X of rolling a dice, the PDF p.x/ is:
p
.
x
/
D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
1=6 x D 1
1=6 x D 2
1=6 x D 3
1=6 x D 4
1=6 x D 5
1=6 x D 6:
Per Equation (2.44), the mean of this discrete random variable of rolling dice is
x
D E
.
x
/
D
X
All i
x
i
p
.
x
i
/
D 1
1
6
C 2
1
6
C 3
1
6
C 4
1
6
C 5
1
6
C 6
1
6
D 3:5:
Example 2.32
e PDF of the Brinell Hardness X of a component is
f
.
x
/
D
8
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
:
0 x < 300
1
70
300 x 370
0 x > 370:
Determine the mean of the Brinell Hardness of the component.
Solution:
Per Equation (2.43), the mean of the Brinell Hardness of this component is:
x
D E
.
x
/
D
Z
1
0
xf
.
x
/
dx D
Z
300
0
x 0dx C
Z
370
300
x
1
70
dx C
Z
1
370
x 0dx
D 0 C
1
140
x
2
ˇ
ˇ
ˇ
ˇ
370
300
C 0 D 335 .Hb/:
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