L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 1 MER301: Engineering...

L Berkley DavisCopyright 2009

MER301: Engineering ReliabilityLecture 3

1

MER301: Engineering Reliability

LECTURE 3:

Random variables and Continuous Random Variables, and Normal Distributions



2

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



4

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



5

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


Probability Density Functions pdf’s describe the set of probabilities associated with

possible values of a random variable X


6

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



7

Histogram Approximation of Probability Density Functions

xall

dxxfxf 1)(1)(

Ax

b

a

dxxfbxaPxfAXP )(][)(][



8

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


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



16

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…



18

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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0.194894



22




23




24




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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


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.



31

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 1 MER301: Engineering...

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