2.9. PROBABILITY FUNCTIONS 55
1. Determine the CDF.
2. Plot the PDF and the CDF.
3. If a component with an inner diameter larger than 1.010 will not be accepted, determine
the percentage of the components that will be discarded.
Solution:
1. e CDF.
Per Equation (2.37), we have the following.
For x < 1:000
00
,
F
.
x
/
D
Z
x
0
f
.
x
/
dx D
Z
x
0
0 dx D 0:
For x 1:000
00
,
F
.
x
/
D
Z
x
0
f
.
x
/
dx D
Z
1:000
0
f
.
x
/
dx C
Z
x
1:000
f
.
x
/
dx
D
Z
1:000
0
0 dx C
Z
x
1:000
401e
401.x1:000/
dx D 1 e
401.x1:000/
:
2. Plots of the PDF and the CDF.
e plots of the PDF and the CDF are shown in Figures 2.12 and 2.13, respectively.
3. P .x > 1:010
00
/.
Per the CDF obtained in step (1) of this example, the probability P .x > 1:010
00
/ is:
P
.
x > 1:010
/
D 1 P
.
x 1:010
/
D 1 F
.
1:010
/
D 1
h
1 e
401
.
x1:000
/
i
D 1 0:9819 D 0:0181 D 1:81%:
e meaning of the probability P.x > 1:010
00
/ D 1:81% is that 1.81% of manufactured
components will be rejected and discarded in this component mass production.
2.9.2 PROBABILITY FUNCTIONS OF A DISCRETE RANDOM
VARIABLE
For a discrete random variable such as rolling a dice, the PDF does not exist because there is a
probability only at discrete points. In this case, we can use the PMF to describe the probability
of a discrete random variable. e CDF is still applicable to a discrete random variable, but the
formula is changed.
56 2. FUNDAMENTAL RELIABILITY MATHEMATICS
f(x)-PDF
500
400
300
200
100
0
0.99 0.995 1 1.005 1.01 1.015 1.02 1.025
x-Inner Diameter (inch)
Figure 2.12: e PDF of the inner diameter.
1
0.75
0.5
0.25
0
0.99 1 1.01 1.02
x-Inner Diameter (inch)
F (x)-CDF
Figure 2.13: e CDF of the inner diameter.
Probability Mass Function (PMF): Probability mass function (PMF) of a discrete random
variable X is a function, denoted as p
.
x
i
/
, that specifies the probability at a discrete point x
i
:
p
.
x
i
/
D P
.
X D x
i
/
; (2.42)
where P .X D x
i
/ is the probability at the point X D x
i
.
Cumulative distribution function (CDF): e CDF F .x/ of a discrete random variable X
is the probability that X will take a value less than or equal to x:
F
.
x
/
D P
.
X x
/
D
X
i
p
.
x
i
/
x x
i
; (2.43)
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