L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 1 MER301: Engineering Reliability LECTURE 3: Random variables and Continuous Random Variables, and Normal Distributions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 2 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 3 Random Variables and Random Experiments  Random Experiment An experiment that can result in different outcomes when repeated in the same manner

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 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

L Berkley Davis Copyright 2009  Probability Density Functions pdf’s describe the set of probabilities associated with possible values of a random variable X MER301: Engineering Reliability Lecture 3 6 Histogram Approximation of Probability Density Functions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 7 Histogram Approximation of Probability Density Functions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 8 Continuous Distribution Probability Density Function

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 9 Cumulative Distribution Function of Continuous Random Variables 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 10 Understanding the Limits of a Continuous Distribution

L Berkley Davis Copyright 2009 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 MER301: Engineering Reliability Lecture 3 12 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 13 Mean and Variance for a Continuous Distribution

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 14 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 15 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 16 Normal Random Variable

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 17 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 18 Characteristics of a Normal Distribution 2.14% 13.6% 34.1% 2.14% 13.6% 34.1%

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 19 Normal Distributions

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 20 Standard Normal Random Variable

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 Standard Normal Random Variable

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 25 Converting a Random Variable to a Standard Normal Random Variable

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 26 Probabilities of Standard Normal Random Variables

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 27 Normal Converted to Standard Normal

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 28 Conversion of Probabilities

L Berkley Davis Copyright 2009 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 3 30 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 31 Summary of Topics  Random Variables  Probability Density and Cumulative Distribution Functions of Continuous Variables  Mean and Variance of Continuous Variables  Normal Distribution