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Engineering Probability and
Statistics - SE-205 -Chap 4
By
S. O. Duffuaa
Loading
Introduction to Probability Density
Function
y
x
(1) F ( x) 
x
Density function of loading on a long, thin beam
f(x)
Introduction to Probability Density
Function
y
x
P(a < X < b)
a
(1) F ( x) 
b
Density function of loading on a long, thin beam
x
Probability Density Function
For a continuous random variable X, a probability density function is a function
such that
(1) f ( x)  0

(2)
 f ( x)dx  1

b
(3) P(a  X  b)   f ( x)dx  area under f ( x) from a to b for any a and b
a
Probability for Continuous Random
Variable
If X is a continuous variable, then for any x1 and x2,
P( x1  X  x2)  P( x1  X  x2 )  P( x1  X  x2 )  P( x1  X  x2 )
Example
Let the continuous random variable X denote the diameter of a hole drilled in a sheet metal component.
The target diameter is 12.5 millimeters. Most random disturbances to the process result in larger
diameters. Historical data show that the distribution of X can be modified by a probability density
function f(x) = 20e-20(x-12.5) , x  12.5.
If a part with a diameter larger than 12.60 millimeters is scrapped, what proportion of parts is scrapped
? A part is scrapped if X  12.60. Now,

P( X  12.60) 


f ( x)dx 
12.6
 20( x 12.5 )
 20( x 12.5 ) 
20
e
dx


e
|12.6  0.135

12.6
What proportion of parts is between 12.5 and 12.6 millimeters ? Now,
12.6
P(12.5  X  12.6) 

.6
f ( x)dx  e 20( x 12.5) |12
12.5  0.865
12.5
Because the total area under f(x) equals one, we can also calculate P(12.5<X<12.6) = 1 – P(X>12.6) =
1 – 0.135 = 0.865
Cumulative Distribution Function
The cumulative distribution function of a continuous random variable X is
x
F ( x)  P( X  x) 
 f (u)du

for    x  
Example for Cumulative Distribution
Function
For the copper current measurement in Example 5-1, the cumulative distribution function of the
random variable X consists of three expressions. If x < 0, then f(x) = 0. Therefore,
F(x) = 0,
for x < 0
x
F ( x)   f (u )du  0.05 x ,
Finally,
for 0  x  20
0
x
F ( x)   f (u )du  1,
for 20  x
0
Therefore,
0

F ( x)  0.05 x
1

The plot of F(x) is shown in Fig. 5-6
x0
0  x  20
20  x
Mean and Variance for Continuous
Random Variable
Suppose X is a continuous random variable with probability density function f(x). The mean or
expected value of X, denoted as  or E(X), is

  E ( X )   xf ( x)dx

The variance of X, denoted as V(X) or 2, is




 2  V ( X )   ( x   ) 2 f ( x)dx   x 2 f ( x)dx   2
The standard deviation of X is  = [V(X)]1/2
Uniform Distribution
A continuous random variable X with probability density function
f ( x)  1 /(b  a) ,
has a continuous uniform distribution
a xb
Uniform Distribution
The mean and variance of a continuous uniform random variable X over a  x  b are
  E ( X )  (a  b) / 2
and
 2  V ( X )  (b  a) 2 / 12
Applications:
• Generating random sample
• Generating random variable
Normal Distribution
A random variable X with probability density function
1
f ( x) 
e
2 
( x )2
2 2
for    x  
has a normal distribution with parameters , where - <  <  , and  > 0. Also,
E ( X )   and V ( X )   2
Normal Distribution
P(     X     )
 0.6827
P(   2  X    2 )  0.9545
P(   3  X    3 )  0.9973
f(x)
 - 3
 - 2
-
-
 - 2
68%
95%
99.7%
Probabilities associated with normal distribution
 - 3
x
Standard Normal
A normal random variable with  = 0 and 2 = 1 is called a standard normal random
variable. A standard normal random variable is denoted as Z.
The cumulative distribution function of a standard normal random variable is denoted as
( z )  P( Z  z )
Standardization
If X is a normal random variable with E(X) =  and V(X) = 2, then the random variable
Z
X 

is a normal random variable with E(Z) = 0 and V(Z) = 1. That is , Z is a standard normal
random variable.
Standardization
Suppose X is a normal random variable with mean  and variance 2 . Then,
 X  x 
P ( X  x )  P

  P( Z  z )
 
 
where,
Z is a standard normal random variable, and
z = (x -  )/ is the z-value obtained by standardizing X.
The probability is obtained by entering Appendix Table II with z = (x -  )/.
Applications:
• Modeling errors
• Modeling grades
• Modeling averages
Binomial Approximation
If X is a binomial random variable, then
X  np
Z
np(1  p)
is approximately a standard normal random variable. The approximation is good for
np > 5
and
n(1-p) > 5
Poisson Approximation
If X is a Poisson random variable with E(X) =  and V(X) = , then
Z
X 

is approximately a standard normal random variable. The approximation is good for
>5
Do not forget correction for
continuity
Exponential Distribution
The random variable X that equals the distance between successive counts of a Poisson
process with mean  > 0 has an exponential distribution with parameter . The probability
density function of X is
f ( x)  ex ,
for 0  x  
If the random variable X has an exponential distribution with parameter  , then
E(X) = 1/  and
V(X) = 1/ 2
Lack of Memory Property
For an exponential random variable X,
P( X  t1  t2 | X  t1 )  P( X  t2 )
Applications:
• Models random time between failures
• Models inter-arrival times between customers
Erlang Distribution
The random variable X that equals the interval length until r failures occur in a Poisson
process with mean  > 0 has an Erlang distribution with parameters  and r. The
probability density function of X is
f ( x) 
r x r 1e x
(r  1)!
,
for x  0 and r  1,2, ...
Erlang Distribution
If X is an Erlang random variable with parameters  and r, then the mean and variance of
X are
 = E(X) = r/  and
2 = V(X) = r/ 2
Applications:
• Models natural phenomena such as rainfall.
• Time to complete a task
Gamma Function
The gamma function is

(r )   x r 1e  x dx,
0
for r  0
Gamma Distribution
The random variable X with probability density function
f ( x) 
r x r 1e x
( r )
,
for x  0
has a gamma distribution with parameters  > 0 and r > 0. If r is an integer, then X has
an Erlang distribution.
Gamma Distribution
If X is a gamma random variable with parameters  and r, then the mean and variance of
X are
 = E(X) = r/  and
2 = V(X) = r/ 2
Applications:
• Models natural phenomena such as rainfall.
• Time to complete a task
Weibull Distribution
The random variable X with probability density function

f ( x) 

 x
 
 
 1
e
( x /  ) 
,
for x  0
has a Weibull distribution with scale parameters  > 0 and shape parameter  > 0
Applications:
• Time to failure for mechanical systems
• Time to complete a task.
Weibull Distribution
If X has a Weibull distribution with parameters  and , then the cumulative distribution
function of X is
F ( x)  1  e
x
 
 

If X has a Weibull distribution with parameters  and , then the mean and variance of x are
 1
   1  
 
and
  1 

2
2
   1     1  
 
   
2
2
2