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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 3 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 4 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 7 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 9 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 10 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 13 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 14 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 15 Normal Random Variable L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 16 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 17 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 18 Normal Distributions L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 19 Standard Normal Random Variable L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 20 Standard Normal Random Variable 0.194894 L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 21 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 23 Standard Normal Random Variable L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 24 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 26 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