1. MATH408: Probability & Statistics Summer 1999 WEEK 3 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu Homepage: www.kettering.edu/~schakrav

2. STUDY OF RANDOM VARIABLES Probability functions Probability density function (continuous) Probability mass function (discrete) Cumulative probability distribution function

3. Probability Density Function

4. Example 3.1 Uniform X = current measured in a thin copper wire (in mA) The PDF of X is given by f(x) = 0.05, 0  x  20.

5. Example 3.1 (cont’d) Find –P( X < 8) –P( X 6 )

6. Example 3.2 Exponential X = diameter (mm) of hole drilled in a sheet metal component

7. Example 3.2 (cont’d) Find –P( X > 12.6) –P( X 12.6 )

8. 3.4.2 Mean and Variance of a Continuous Random Variable (page 61)

9. EXAMPLES

10. NORMAL (GAUSSIAN) The most important continuous distribution in probability and statistics The story of the outcome of normal is really the story of the development of statistics as a science. Gauss discovered this while incorporating the method of least squares for reducing the errors in fitting curves for astronomical observations.

11. PDF OF NORMAL

12. Graphs of various normal PDF

15. ILLUSTRATION OF CALCULATION OF NORMAL PROBABILITIES- EXAMPLE 3.7

16. EXAMPLE (cont’d)

18. How to standardize?

19. Standardize (cont’d)

20. EXAMPLES

21. HOME WORK PROBLEMS CHAPTER 3 Sections: 3.1 through 3.5 1-9, 10, 15, 18, 21, 22, 23, 27-29, 31

22. Probability Plots (revisiting) Frequently you will be dealing with the assumption of normal populations. Questions: (1) How do we verify this? (2) How do we rectify if the assumption is violated? To answer (1), we look at probability plot (normal probability plot). For (2), we use transformation.

23. Plots(cont’d) Construction of a probability plot can be done in two ways. First calculate the percentiles of the data points, say x (j). 1. On the probability paper (which will have the percentiles along the y-axis and the values of the data along the x- axis) plot the values, x (j),.

24. Plots(cont’d) 2. Calculate y (j), the corresponding percentiles of the probability distribution. Plot (x (j),y (j), ) on a regular paper. If the points lie pretty much on a straight line, then we can conclude that there is no evidence to refute the assumption.