Engineering Probability and Statistics - SE-205 -Chap 4

1
Engineering Probability and Statistics - SE-205
-Chap 4

• By
• S. O. Duffuaa

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Introduction to Probability Density Function
Loading
x
Density function of loading on a long, thin beam
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Introduction to Probability Density Function
f(x)
P(a lt X lt b)
a
b
x
Density function of loading on a long, thin beam
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Probability Density Function
For a continuous random variable X, a probability
density function is a function such that
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Probability for Continuous Random Variable
If X is a continuous variable, then for any x1
and x2,
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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,
What proportion of parts is between 12.5 and 12.6
millimeters ? Now,
Because the total area under f(x) equals one, we
can also calculate P(12.5ltXlt12.6) 1 P(Xgt12.6)
1 0.135 0.865
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Cumulative Distribution Function
The cumulative distribution function of a
continuous random variable X is
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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 lt 0, then f(x) 0. Therefore, F(x) 0,
for x lt 0
Finally,
Therefore,
The plot of F(x) is shown in Fig. 5-6
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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
The variance of X, denoted as V(X) or ?2, is
The standard deviation of X is ? V(X)1/2
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Uniform Distribution
A continuous random variable X with probability
density function
has a continuous uniform distribution
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Uniform Distribution
The mean and variance of a continuous uniform
random variable X over a ? x ? b are

• Applications
• Generating random sample
• Generating random variable

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Normal Distribution
A random variable X with probability density
function
has a normal distribution with parameters ?,
where -? lt ? lt ? , and ? gt 0. Also,
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Normal Distribution
f(x)
? - 3? ? - 2? ? - ?
? - ? ? - 2? ? - 3?
x
68
95
99.7
Probabilities associated with normal distribution
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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
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Standardization
If X is a normal random variable with E(X) ?
and V(X) ?2, then the random variable
is a normal random variable with E(Z) 0 and
V(Z) 1. That is , Z is a standard normal random
variable.
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Standardization
Suppose X is a normal random variable with mean ?
and variance ?2 . Then,
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

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Binomial Approximation
If X is a binomial random variable, then
is approximately a standard normal random
variable. The approximation is good for
np gt 5
and n(1-p) gt 5
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Poisson Approximation
If X is a Poisson random variable with E(X) ?
and V(X) ?, then
is approximately a standard normal random
variable. The approximation is good for

? gt 5
Do not forget correction for continuity
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Exponential Distribution
The random variable X that equals the distance
between successive counts of a Poisson process
with mean ? gt 0 has an exponential distribution
with parameter ?. The probability density
function of X is
If the random variable X has an exponential
distribution with parameter ? , then
E(X) 1/ ?
and V(X) 1/ ?2
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Lack of Memory Property
For an exponential random variable X,

• Applications
• Models random time between failures
• Models inter-arrival times between customers

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Erlang Distribution
The random variable X that equals the interval
length until r failures occur in a Poisson
process with mean ? gt 0 has an Erlang
distribution with parameters ? and r. The
probability density function of X is
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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

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Gamma Function
The gamma function is
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Gamma Distribution
The random variable X with probability density
function
has a gamma distribution with parameters ? gt 0
and r gt 0. If r is an integer, then X has an
Erlang distribution.
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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

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Weibull Distribution
The random variable X with probability density
function
has a Weibull distribution with scale parameters
? gt 0 and shape parameter ? gt 0

• Applications
• Time to failure for mechanical systems
• Time to complete a task.

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Weibull Distribution
If X has a Weibull distribution with parameters ?
and ?, then the cumulative distribution function
of X is
If X has a Weibull distribution with parameters ?
and ?, then the mean and variance of x are
and