1. COMP 170 L2 L17: Random Variables and Expectation Page 1

2. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties n Expectation and counting n Geometric distribution Page 2

3. COMP 170 L2 Functions on Sample Space l Probability starts with a process (test, experiment) whose outcome is uncertain l Sample space: the set of all possible outcomes l Sometimes, we want to define functions on the sample space l Example n Process: flip a coin n times n Sample space: set of sequences of n elements, each being H or T n Example: n=5, Function: “Number of heads” Page 3

4. COMP 170 L2 Random Variables l Probability starts with a process whose outcome is uncertain l Sample space: the set of all possible outcomes l Sometimes, we want to define functions on the sample space l Outcome of the process is uncertain l Function defined on the outcome is also uncertain l So, called random variable l A random variable is a function defined on the sample space l Example: “Number of heads” in n coin flips is a random variables Page 4

5. COMP 170 L2 Example 2 Page 5

6. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties n Expectation and counting n Geometric distribution Page 6

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12. COMP 170 L2 Binomial Random Variable l n Bernoulli trials, each with probability of success at each trial being p l X: number of successes, l X: called Binomial random variable l P: Binomial probability distribution l Do the numbers sum to 1? Page 12

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14. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties n Expectation and counting n Geometric distribution Page 14

15. COMP 170 L2 Expectation of Random Variables Page 15

16. COMP 170 L2 Expectation Example Page 16

17. COMP 170 L2 Expectation and Average l Expectation ~= Average over many runs of process Page 17 Expect the average

18. COMP 170 L2

19. Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties  Proof of linearity  Use of linearity n Expectation and counting n Geometric distribution Page 19

20. COMP 170 L2 Expectation from Sample space Page 20 l Get the same result from sample space

21. COMP 170 L2 Expectation from Sample space Page 21

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27. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties  Proof of linearity  Use of linearity n Expectation and counting n Geometric distribution Page 27

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33. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties n Expectation and counting n Geometric distribution Page 33

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38. COMP 170 L2 Outline l Random Variables n The concept n Bernoulli/Binomial random variables l Expectation of Random Variables n The concept n Properties n Expectation and counting n Geometric distribution Page 38

39. COMP 170 L2 Repeating Bernoulli Trials Until Success Page 39

40. COMP 170 L2 Number of Trials until First Success Page 40

41. COMP 170 L2 l Let x = 1-p l Consider only 0 < p < 1 l Then, 0 < x <1 l Let n goes to infinity, LHS becomes Page 41

42. COMP 170 L2 Number of Trial until First Success Page 42

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45. COMP 170 L2 Recap: 06-05-2010 l Probability starts with a process whose outcome is uncertain l Sample space: the set of all possible outcomes l A random variable is a function defined on the sample space n Uncertain because outcome of process is l Bernoulli RV: n Outcome of Bernoulli Trail (success of failure) l Binomial RV n # of successes in n independent Bernoulli trials