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MER301: Engineering
Reliability
LECTURE 3:
Random variables and Continuous
Random Variables, and Normal
Distributions
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
1
Summary of Topics
 Random Variables
 Probability Density and Cumulative
Distribution Functions of Continuous
Variables
 Mean and Variance of Continuous
Variables
 Normal Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
2
Random Variables and Random Experiments
 Random Experiment

L Berkley Davis
Copyright 2009
An experiment that can result in different outcomes
when repeated in the same manner
MER301: Engineering Reliability
Lecture 3
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Random Variables
 Random Variables
 Discrete
 Continuous
 Variable Name Convention
 Upper case
 Lower case
X
x
X &x
the random variable
a specific numerical value
Random Variables are Characterized by a Mean and a Variance
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Calculation of Probabilities
 Probability Density Functions
 pdf’s describe the set of probabilities
associated with possible values of a
random variable X
 Cumulative Distribution Functions
 cdf’s describe the probability, for a given
pdf, that a random variable X is less than
or equal to some specific value x
cdf  P ( X  x)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
5
Histogram Approximation of
Probability Density Functions
 Probability Density Functions
Data
Xi
68.4
66.4
69.5
 pdf’s describe the set of probabilities associated
with possible values of a random variable X
Descriptive Statistics
71.6
66.6
Variable: Xi
72.5
69.6
Anderson-Darling Normality Test
68.5
71.2
66.8
70.3
65.6
65.3
67.1
68.9
64.8
67.9
70.3
69.1
67.8
67.4
A-Squared:
P-Value:
64.5
66.5
68.5
70.5
Minimum
1st Quartile
Median
3rd Quartile
Maximum
95% Confidence Interval for Mu
68.4920
1.9820
3.92827
2.91E-02
-4.6E-01
25
64.8000
66.9500
68.5000
69.9500
72.5000
95% Confidence Interval for Mu
67.6739
67.5
68.5
69.5
69.3101
95% Confidence Interval for Sigma
1.5476
70.5
69.3
72.5
Mean
StDev
Variance
Skewness
Kurtosis
N
0.084
0.998
2.7572
95% Confidence Interval for Median
95% Confidence Interval for Median
67.4792
69.4604
68.8
68.1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
6
Histogram Approximation of
Probability Density Functions
 f ( x)  1


 f ( x )  dx  1

all x
b
P[ X  A]   f ( x )  P[a  x  b]   f ( x )  dx
x A
L Berkley Davis
Copyright 2009
a
MER301: Engineering Reliability
Lecture 3
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Continuous Distribution Probability
Density Function
f ( x)  0

 f ( x)  dx  1

a
b


a
b
 f ( x)  dx   f ( x)  dx   f ( x)  dx  1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
8
Cumulative Distribution Function
of Continuous Random Variables
Cumulative Distribution Function
1.2
1
CDF
0.8
0.6
c
0.4
0.2
0
-10
-8
-6
-4
-2
0
2
4
6
X
Graphically this probability corresponds to the area under
The graph of the density to the left of and including x
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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8
10
Understanding the Limits of a
Continuous Distribution
x
P( X  x)   f ( x)  dx  0
x
P( x1  X  x2 )  P( x1  X  x2 ) 
x2
 f ( x)  dx
x1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Example 3.1
 The concentration of vanadium,a corrosive
metal, in distillate oil ranges from 0.1 to 0.5
parts per million (ppm).
 The Probability Density Function is given by
 f(x)=12.5x-1.25, 0.1 ≤ x ≤ 0.5
 0 elsewhere
 Show that this is in fact a pdf
 What is the probability that the vanadium
concentration in a randomly selected sample of
distillate oil will lie between 0.2 and 0.3 ppm.
L Berkley Davis
Copyright 2009
Example 3.2
 The density function for the Random
Variable x is given in Example 3.1
 Determine the cumulative distribution
function F(x)
 What is F(x) in the given range of x
 x<0.1
 0.1<x<0.5
 x>0.5
 Use the cumulative distribution function to
calculate the probability that the vanadium
concentration is less than 0.3ppm
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
12
Mean and Variance for a
Continuous Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Example 3.3
 Determine the Mean, Variance,
and Standard Deviation for the
density function of Example 3.1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Normal Distribution
 Many Physical Phenomena are
characterized by normally distributed
variables
 Engineering Examples include variation
in such areas as:




L Berkley Davis
Copyright 2009
Dimensions of parts
Experimental measurements
Power output of turbines
Material properties
MER301: Engineering Reliability
Lecture 3
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Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Characteristics of a Normal
Distribution
Symmetric bell shaped curve
Centered at the Mean
Points of inflection at µ±σ
A Normally Distributed Random
Variable must be able to assume any
value along the line of real numbers
 Samples from truly normal
distributions rarely contain outliers…




L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Characteristics of a Normal
Distribution
f ( x) 
d2 f
 0 @ x  1
dx 2
34.1%
34.1%
13.6%
13.6%
2.14%
L Berkley Davis
Copyright 2009
1
@x  
2 
2.14%
MER301: Engineering Reliability
Lecture 3
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Normal Distributions
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Standard Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Standard Normal Random Variable
0.194894
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Standard Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
22
Standard Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Standard Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Converting a Random Variable to a
Standard Normal Random Variable
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
25
Probabilities of Standard Normal
Random Variables
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
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Normal Converted to Standard Normal
 2
  10
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
Z
X 


X  10
2
27
Conversion of Probabilities
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
28
Normal Distribution in Excel
NORMDIST(x,mean,standard_dev,cumulative)
X is the value for which you want the distribution.
Mean is the arithmetic mean of the distribution.
Standard_dev is the standard deviation of the distribution.
Cumulative is a logical value that determines the form of the function. If
cumulative is TRUE, NORMDIST returns the cumulative distribution
function; if FALSE, it returns the probability mass function.
Remarks
If mean or standard_dev is nonnumeric, NORMDIST returns the #VALUE! error value.
If standard_dev ≤ 0, NORMDIST returns the #NUM! error value.
If mean = 0 and standard_dev = 1, NORMDIST returns the standard normal distribution,
NORMSDIST.
Example
=NORMDIST(42,40,1.5,TRUE) equals 0.908789
L Berkley Davis
Copyright 2009
Example 3.4
 Let X denote the number of grams of
hydrocarbons emitted by an automobile
per mile.
 Assume that X is normally distributed
with a mean equal to 1 gram and with a
standard deviation equal to 0.25 grams
 Find the probability that a randomly
selected automobile will emit between
0.9 and 1.54 g of hydrocarbons per mile.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
30
Summary of Topics
 Random Variables
 Probability Density and Cumulative
Distribution Functions of Continuous
Variables
 Mean and Variance of Continuous
Variables
 Normal Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 3
31