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: x0123x0123 P(x) 1/8 3/8 1/8 P(x) 0 1 2 3 1/8 2/8 3/8

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

5. P(X=1) = 3/8 H H H H H H H T T T T T 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) =  =  x i P(x i )

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 Success Failure.3.7 Good Bad Good Bad.9.1.9.1.27.03.63.07

11. X = 1010 P(X = 1) = P(A) inherits probabilities Bernoulli RV

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

15. Variance of a Bernoulli RV V(X) = p - p 2 = 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 P(Y = k) =p k (1-p) n-k ; k = 0,1,..., n Count occurrences in n trials 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 xP(x) 00.006047 10.040311 20.120932 30.214991 40.250823 50.200658 60.111477 70.042467 80.010617 90.001573 100.000105

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

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