l berkley davis copyright 2009 mer301: engineering reliability lecture 3 1 mer301: engineering...
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L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 3
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MER301: Engineering Reliability
LECTURE 3:
Random variables and Continuous Random Variables, and Normal Distributions
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Summary of Topics
Random Variables Probability Density and Cumulative
Distribution Functions of Continuous Variables
Mean and Variance of Continuous Variables
Normal Distribution
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Random Variables and Random Experiments
Random Experiment An experiment that can result in different outcomes when
repeated in the same manner
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Random Variables
Random Variables Discrete Continuous
Variable Name Convention Upper case the random variable Lower case a specific numerical value
Random Variables are Characterized by a Mean and a Variance
xX &Xx
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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
)( xXPcdf
L Berkley DavisCopyright 2009
Probability Density Functions pdf’s describe the set of probabilities associated with
possible values of a random variable X
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Histogram Approximation of Probability Density Functions
DataXi
68.466.4
69.5
71.6
66.6
72.5
69.6
68.5
71.266.870.365.6
65.3
67.168.964.867.970.3
69.167.8
67.4
70.5
69.3
68.868.1
72.570.568.566.564.5
95% Confidence Interval for Mu
69.568.567.5
95% Confidence Interval for Median
Variable: Xi
67.4792
1.5476
67.6739
Maximum3rd QuartileMedian1st QuartileMinimum
NKurtosisSkewnessVarianceStDevMean
P-Value:A-Squared:
69.4604
2.7572
69.3101
72.500069.950068.500066.950064.8000
25-4.6E-012.91E-023.92827 1.982068.4920
0.9980.084
95% Confidence Interval for Median
95% Confidence Interval for Sigma
95% Confidence Interval for Mu
Anderson-Darling Normality Test
Descriptive Statistics
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Histogram Approximation of Probability Density Functions
xall
dxxfxf 1)(1)(
Ax
b
a
dxxfbxaPxfAXP )(][)(][
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Continuous Distribution Probability Density Function
1)()()(
1)(
0)(
b
a b
a
dxxfdxxfdxxf
dxxf
xf
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Cumulative Distribution Functionof Continuous Random Variables
Graphically this probability corresponds to the area underThe graph of the density to the left of and including x
Cumulative Distribution Function
0
0.2
0.4
0.6
0.8
1
1.2
-10 -8 -6 -4 -2 0 2 4 6 8 10
X
CD
F
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Understanding the Limits of aContinuous Distribution
2
1
)()()( 2121
x
x
dxxfxXxPxXxP
0)()( x
x
dxxfxXP
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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.
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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
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Mean and Variance for a Continuous Distribution
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Example 3.3
Determine the Mean, Variance, and Standard Deviation for the density function of Example 3.1
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Normal Distribution Many Physical Phenomena are
characterized by normally distributed variables
Engineering Examples include variation in such areas as: Dimensions of parts Experimental measurements Power output of turbines Material properties
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Normal Random Variable
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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…
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Characteristics of a Normal Distribution
2.14%
13.6%
34.1%
2.14%
13.6%
34.1%
xxf @2
1)(1@0
2
2
xdx
fd
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Normal Distributions
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Standard Normal Random Variable
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Standard Normal Random Variable
0.194894
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Standard Normal Random Variable
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Standard Normal Random Variable
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Standard Normal Random Variable
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Converting a Random Variable to a Standard Normal Random Variable
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Probabilities of Standard Normal Random Variables
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Normal Converted to Standard Normal
2
10
XXZ
10
2
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Conversion of Probabilities
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Normal Distribution in ExcelNORMDIST(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
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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.
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Summary of Topics
Random Variables Probability Density and Cumulative
Distribution Functions of Continuous Variables
Mean and Variance of Continuous Variables
Normal Distribution