Lecture 8. Discrete Probability Distributions

1. Lecture 8. Discrete Probability Distributions David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management

2. AGENDA • Review Random Variables • Binomial Process • Binomial Distribution • Poisson Process • Poisson Distribution

3. Example 1. Flip Three Coins • Sample space is HHH, HHT, HTH, THH, HTT, THT, TTH, TTT • X = the number of heads, X = 0, 1, 2, 3 • Probability Distribution: P(x) x 0 1 2 3 P(x) 1/8 3/8 3/8 1/8 3/8 2/8 1/8 0 1 2 3

4. Probability Tree Form 1/8 H H 1/8 T H 1/8 H T 1/8 T H 1/8 H T T 1/8 H 1/8 T T 1/8

5. P(X=1) = 3/8 1/8 H H 1/8 T H 1/8 H T 1/8 T H 1/8 H T T 1/8 H 1/8 T T 1/8

6. Expected Value • The expected value (mean) of a probability distribution is a weighted average: weights are the probabilities • Expected Value: E(X) =  = xiP(xi)

7. Calculating Expected Value E(X) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 1.5

8. Variance V(X) = E(X-)2

9. Calculating Variance: 

10. Example 2. Program Pilot--Bayes Rule .27 Good .9 .1 Success Bad .03 .3 .63 .7 Good Failure .9 .1 Bad .07

11. Bernoulli RV 1 0 X = P(X = 1) = P(A) inherits probabilities

12. Application: Survey of Employment Discrimination • Wall Street Journal, 1991 May 15 • Pairs of equally qualified white and black applicants for entry-level positions • Dichotomy: job offer or not • Results: • 28% of whites offered jobs • 18% of blacks offered jobs

13. Bernoulli Probabilities

14. Expected Value 1-p p 0 1 note long-run relative frequency interpretation

15. Variance of a Bernoulli RV V(X) = p - p2 = p(1-p)

16. Bernoulli Process 1 2 3 4 5 6 A sequence of independent Bernoulli trials each with probability p of taking on the value 1 Application: Examine “abandoned” buildings to see if they are in fact occupied

17. Binomial Distribution Count occurrences in n trials P(Y = k) = pk(1-p)n-k; k = 0,1,..., n Survey 1200 buildings. How many are actually occupied?

18. Parameters • Mean:  = n p • Variance:  = n p q • Standard Deviation: 

19. Example 3. Racial Discrimination • Stermerville Public Works Department charged with racial discrimination in hiring practices • 40% of the persons who passed the department’s civil service exam were minorities • From this group, the Department hired 10 individuals; 2 of them were minorities. • What is the probability that, if the Department did not discriminate, it would have hired 2 or fewer minorities?

20. Example 3. Solution • Success: a minority is hired • Probability of success: p = 0.4, if the department shows no preferences in regard to hiring minorities • Number of trials, n = 10 • Number of successes, x = 2 • P(x  2) = 0.12 + 0.04 + 0.006 = 0.166 .

21. Example 4. Probability Distribution x P(x) 0 0.006047 1 0.040311 2 0.120932 3 0.214991 4 0.250823 5 0.200658 6 0.111477 7 0.042467 8 0.010617 9 0.001573 10 0.000105

22. Poisson Process rate x x x time 0 Assumptions time homogeneity independence no clumping

23. Application: Toll Booth • Arrival times of cars • Mean arrival rate,  cars per minute • Busy:cars per minute • Slow: = 0.5 cars per minute

24. Poisson Distribution Count in time period t

25. Probability Calculation

26. Poisson Mean and Variance

27. Next Time ... • Continuous Probability Distributions • Normal Distribution