1. PROBABILITY AND STATISTICS ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier

2. Copyright © 1999 by Lizette R. Chevalier Permission is granted to students at Southern Illinois University at Carbondale to make one copy of this material for use in the class ENGR 351, Numerical Methods for Engineers. No other permission is granted. All other rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner.

3. Why statistics in a course on numerical methods? Inputs on models are typically variable and uncertain –media specific hydraulic conductivity organic carbon content wind speed –receptor specific body weight ingestion rate –chemical specific Henry’s law constant decay rate toxicity values –landuse specific industrial commercial

4. Why statistics in a course on numerical methods? Inputs on models are typically variable and uncertain Best represented by probability distributions Tools for estimating values Quantifying uncertainty Tool for risk assessment

5. Random Variables Discrete – variable can only attain a finite number of values. Continuous –variable can be from zero to infinity.

6. Probability

7. The following properties are associated with probability. 1. Probability is a nonnegative number. 2. If an event is certain, then m and n are equal and p(x) = 1 3. Mutually exclusive p(A+B) = p(A) + p(B) In other words, the probability of A or B is the sum of the probability of either event. For example, the probability of 2 or 4 on the throw of a dice is: p(2) + p(4) = 1/6 + 1/6 = 1/3

8. 4. If events are independent p(AB) = p(A). p(B) The occurrence of one does not affect the occurrence of the other. For example, the probability that 2 and 4 will occur in two dice simultaneously thrown is: p(2,4) = p(2). p(4). 2 = (1/6). (1/6). (2) = 1/18

9. Density Function The density function is defined as the function which yields the probability that the random variable takes on any one of its values. Density function for a continuous random variable Probability of Occurrence p(x) = f(x) x

10. # on dice f(x) 1/6 Density function for a discrete random variable

11. Cumulative Distribution Function, F(x) F(x) x Continuous function Allows us to determine the probability that x is less than or equal to a

12. For a continuous variable For a discrete variable

13. F(x) 1 5/6 2/3 1/2 1/3 1/6 0123456 # on dice Cumulative distance: Discrete function

14. which is the area under the density function Probability of Occurrence p(x) = f(x) x a

15. Samples and Populations Sample- a random selection of items from a lot or population in order to evaluate the characteristics of the lot or population –mean –expected value –variance

16. Mean or Expected Value continuous discrete In the case of a discrete sample, is this the mean?

17. Example: Expected value on dice...end of problem

18. Example: Let x = # of hours of a light bulb. Find the expected life.

19. Example (cont.)...end of problem

20. Variance Describes the “spread” or shape of the distribution 01234 123

21. discrete continuous The following equations describe the discrete and continuous cases for variance.

22. Example: Let x = number of orders received per day Probability density function: Company A Probability density function: Company B

23. Probability density function: Company A Probability density function: Company B

24. Variance Company A Company B 01234 123...end of problem

25. Third moment measure of asymmetry +- If symmetric S= 0 Often referred to as skewness, s

26. Fourth moment, measure of flatness small kurtosis large kurtosis

27. Monte Carlo Technique A technique for modeling processes that involve random variables. You need: a random variable and its probability distribution a sequence of random numbers

28. Suppose x is a random number that describes the demand per day of a commodity where:

29. Cumulative distribution function

30. The task is to generate values of x (demand) such that the relative frequency of each value of k will be equal to its probability. Need a sequence of random numbers Will use the “MIDSQUARE” technique

31. Monte Carlo Method 1. Take a 4 digit number (preferably selected at random) 2. Square the number 3. Take 4 digits starting at the third from the left. 4. Record 5. Square -------- etc

32. ie. Select 1653 (1653) 2 = 53640976 Select 6409 1653, 6409, 0752,.........

34. Recall that we are studying the following system: Suppose x is a random number that describes the demand per day of a commodity where:

35. The “middle” digits are considered random. Since the study of demand requires 4 subsets. Using: xRandom 00 - 999(10%) 11000 - 2999(20%) 23000 - 6999(40%) 37000 - 9999(30%) Number 7324x = 3 6409 x = 2

36. xRandom 00 - 999 11000 - 2999 23000 - 6999 37000 - 9999 RandomCt. # 91403 53962 11681 36422 Total: 8

37. Example: Generate numbers with class Probability of hydraulic conductivity Soil samples at site: K(cm/s)f(x) 10 -5 (silty sand)0.4 10 -4 0.35 10 -3 (sand)0.15 10 -2 0.07 10 -1 (gravel)0.03 depth Ground surface

38. Soil samples at site: K(cm/s)f(x)Random # 10 -5 (silty sand)0.40-3999 10 -4 0.354000 - 7499 10 -3 (sand)0.157500 - 8999 10 -2 0.079000 - 9699 10 -1 (gravel)0.039700 - 9999

39. Soil samples at site: K(cm/s)f(x)Random # 10 -5 (silty sand)0.40-3999 10 -4 0.354000 - 7499 10 -3 (sand)0.157500 - 8999 10 -2 0.079000 - 9699 10 -1 (gravel)0.039700 - 9999 NodeRandom #K (cm/s) (1,1) (2,1) (3,1) (4,1) (5,1)

40. end of lecture