2.9. PROBABILITY FUNCTIONS 51
manually determine the lower boundary and upper boundary of each bin and arrange them in
another column; and (3) activate the “Data Analysis,” select “Histogram” and then follow the
pop-up instructions in the prompt window. e “Data Analysis” will automatically count the
frequencies in each bin per Equation (2.32) and create the histogram.
Histogram can be easily created in MATLAB by the following command:
histogram .X; J /;
where “X ” is a matrix which contains all of the sample data of a random variable. “J ” is the num-
ber of bins, which is typically larger than or equal to 6. Execution of the command “histogram
(X, J )” will automatically create a histogram.
2.9 PROBABILITY FUNCTIONS
2.9.1 PROBABILITY FUNCTIONS OF A CONTINUOUS RANDOM
VARIABLE
When the sample size of a continuous random variable is infinite, and the bin width is infinites-
imal, the relative-density frequency, as shown in Figure 2.8 and discussed in the previous section
will be a PDF.
Probability Density Function(PDF): For a continuous random variable X , the function f .x/
is defined as a probability per unit of random variable value, named as PDF if it satisfies the
following three conditions:
f
.
x
/
0 1 < x < 1 (2.34)
P
.
a X b
/
D
Z
b
a
f
.
x
/
dx (2.35)
P
.
1 X 1
/
D
Z
1
1
f
.
x
/
dx D 1: (2.36)
Any function can be used as a PDF of a continuous random variable if it satisfies Equa-
tions (2.34), (2.35), and (2.36).
e physical meaning of f .x/ is the probability per unit of the random variable value,
which is similar to the mass per unit length or the loading per unit length in beam theory. Since
f .x/ is related to probability, a negative value of f .x/ has no physical meaning. erefore, f .x/
must be large than or equal to 0, as shown in Equation (2.34).
Per the definition of
f .x/
, the probability of the event
.x
X
x
C
dx/
is equal to
f .x/dx, that is, P
.
x X x C dx
/
D f
.
x
/
dx. So, the probability of the event .a X b/
will be equal to
R
b
a
f
.
x
/
dx, as shown in Equation (2.35).
Event
.
1 X 1
/
is a universal set of a continuous random variable. erefore, the
probability of the event
.
1 X 1
/
is certainly equal to 1, as shown in Equation (2.36).
52 2. FUNDAMENTAL RELIABILITY MATHEMATICS
We can use the PDF to define another important function: the cumulative distribution
function.
Cumulative Distribution Function (CDF): e CDF F .x/, also known as the PDF, of a
random variable X is the probability that X will take a value less than or equal to x:
F
.
x
/
D P
.
1 X x
/
D
Z
x
1
f
.
x
/
dx; (2.37)
where f
.
x
/
is the PDF of a random variable x.
If a random variable X is continuous and the first derivative of the CDF F .x/ exists, the
PDF f
.
x
/
is equal to the first derivative of F
.
x
/
, that is,
f
.
x
/
D
dF .x/
dx
: (2.38)
Per the definition of the CDF as expressed in Equation (2.36), we can get following several
conclusions about the CDF of a continuous random variable X:
F
.
1
/
D
Z
1
1
f
.
x
/
dx D 0 (2.39)
F
.
1
/
D
Z
1
1
f
.
x
/
dx D 1 (2.40)
P
.
a X b
/
D
Z
b
a
f
.
x
/
dx D
Z
b
1
f
.
x
/
dx
Z
a
1
f
.
x
/
dx
D F
.
b
/
F .a/: (2.41)
Example 2.28
e PDF f
.
x
/
of a continuous random variable with a value in the range of
.
1; 1
/
is shown
in Figure 2.10. (1) Based on the curve, build the PDF of this random variable X. (2) De-
termine and plot the CDF. (3) Use both of the PDF and CDF to calculate the probability
P
.
1:5 X 3:5
/
.
Solution:
1. e PDF of the random variable.
Per the PDF curve provided, we can have the PDF of this random variable:
f
.
x
/
D
8
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
:
0 1 x 1
.
x 1
/
=2 1 x 2
.
5 x
/
=6 2 x 5
0 5 x 1:
2.9. PROBABILITY FUNCTIONS 53
0.6
0.5
0.4
0.3
0.2
0.1
0
-2 -1 0 1 2 3 4 5 6 7 8 9 10
x-Random Variable
f(x)-PDF
Figure 2.10: e curve of the PDF.
2. e CDF of the random variable.
Per Equation (2.36), the CDF of this random variable will be:
F
.
x
/
D
Z
x
1
f
.
x
/
dx D
Z
x
1
0 dx D 0 1 x 1
F
.
x
/
D
Z
x
1
f
.
x
/
dx D
Z
1
1
f
.
x
/
dx C
Z
x
1
f
.
x
/
dx
D
Z
1
1
0 dx C
Z
x
1
1
2
.
x 1
/
dx D
1
2
x
2
2
x C
1
2
1 x 2
F
.
x
/
D
Z
x
1
f
.
x
/
dx D
Z
1
1
f
.
x
/
dx C
Z
2
1
f
.
x
/
dx C
Z
x
2
f
.
x
/
dx
D
Z
1
1
0 dx C
Z
2
1
1
2
.
x 1
/
dx C
Z
x
2
1
6
.
5 x
/
dx
D 0:25 C
1
6
5x
x
2
2
8
2 x 5
F
.
x
/
D
Z
x
1
f
.
x
/
dx D
Z
1
1
f
.
x
/
dx C
Z
2
1
f
.
x
/
dx C
Z
5
2
f
.
x
/
dx C
Z
x
5
f
.
x
/
dx
D
Z
1
1
0 dx C
Z
2
1
1
2
.
x 1
/
dx C
Z
5
2
1
6
.
5 x
/
dx C
Z
x
5
0 dx
D 1 5 x 1:
e CDF of this random variable is plotted in Figure 2.11.
54 2. FUNDAMENTAL RELIABILITY MATHEMATICS
1.2
1
0.8
0.6
0.4
0.2
0
-2 -1 0 1 2 3 4 5 6 7 8 9 10
x-Random Variable
F(x)-CDF
Figure 2.11: e curve of the CDF.
3. e probability P
.
1:5 X 3:5
/
.
e probability P
.
1:5 X 3:5
/
can be determined by using the PDF f
.
x
/
as follows:
P
.
1:5 X 3:5
/
D
Z
2
1:5
f
.
x
/
dx C
Z
3:5
2
f
.
x
/
dx
D
Z
2
1:5
1
2
.
x 1
/
dx C
Z
3:5
2
1
6
.
5 x
/
dx
D 0:1875 C 0:5625 D 0:75:
e probability P
.
1:5 X 3:5
/
can also be directly determined by using the PDF F .x/
as follows:
P
.
1:5 X 3:5
/
D F
.
3:5
/
F
.
1:5
/
D
"
0:25 C
1
6
5 3:5
3:5
2
2
8
!#
"
1
2
1:5
2
2
1:5 C
1
2
!#
D 0:8125 0:0625 D 0:75:
Example 2.29
e PDF of inner diameter X of a component in mass production is:
f
.
x
/
D
8
<
:
0 1:000
00
< x
401e
401.x1:000/
x 1:000
00
:
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