MER301: Engineering Reliability

1. MER301: Engineering Reliability LECTURE 3: Random variables and Continuous Random Variables, and Normal Distributions 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 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 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 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 MER301: Engineering Reliability Lecture 3

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

7. Histogram Approximation of Probability Density Functions MER301: Engineering Reliability Lecture 3

8. Continuous Distribution Probability Density Function MER301: Engineering Reliability Lecture 3

9. Cumulative Distribution Functionof Continuous Random Variables Graphically this probability corresponds to the area under The graph of the density to the left of and including x MER301: Engineering Reliability Lecture 3

10. Understanding the Limits of aContinuous Distribution MER301: Engineering Reliability Lecture 3

11. 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.

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 MER301: Engineering Reliability Lecture 3

13. Mean and Variance for a Continuous Distribution MER301: Engineering Reliability Lecture 3

14. Example 3.3 • Determine the Mean, Variance, and Standard Deviation for the density function of Example 3.1 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 MER301: Engineering Reliability Lecture 3

16. Normal Random Variable 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… MER301: Engineering Reliability Lecture 3

18. Characteristics of a Normal Distribution 34.1% 34.1% 13.6% 13.6% 2.14% 2.14% MER301: Engineering Reliability Lecture 3

19. Normal Distributions MER301: Engineering Reliability Lecture 3

20. Standard Normal Random Variable MER301: Engineering Reliability Lecture 3

21. Standard Normal Random Variable 0.194894 MER301: Engineering Reliability Lecture 3

22. Standard Normal Random Variable MER301: Engineering Reliability Lecture 3

23. Standard Normal Random Variable MER301: Engineering Reliability Lecture 3

24. Standard Normal Random Variable MER301: Engineering Reliability Lecture 3

25. Converting a Random Variable to a Standard Normal Random Variable MER301: Engineering Reliability Lecture 3

26. Probabilities of Standard Normal Random Variables MER301: Engineering Reliability Lecture 3

27. Normal Converted to Standard Normal MER301: Engineering Reliability Lecture 3

28. Conversion of Probabilities MER301: Engineering Reliability Lecture 3

29. Normal Distribution in Excel NORMDIST(x,mean,standard_dev,cumulative)Xis the value for which you want the distribution.Mean is the arithmetic mean of the distribution.Standard_devis the standard deviation of the distribution.Cumulativeis 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.RemarksIf 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

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. 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 MER301: Engineering Reliability Lecture 3 31