1. Binomial and Geometric Distributions Delta On-Time Performance at Hartsfield- Jackson Atlanta International (June, 2003 - June, 2015) http://www.transtats.bts.gov/OT_Delay/ot_delaycause1.asp?display=data&pn=1

2. Data / Model Total Operations: 2,278,897 On-Time Operations: 1,824,432 Proportion On-Time: 1824432/2278897 =.8006 (.80) Will consider random samples of various sizes from this population of operations Y ≡ # of On-Time operations out of the sample of n Y ~ Binomial(n, p Y = 0.80) X ≡ # of Flights sampled until the first NOT On-Time Arrival is selected X ~ Geometric(p X = 0.20)

3. Binomial Distribution – Probability Function

4. Geometric Distribution Used to model the number of Bernoulli trials needed until the first Success occurs (P(S)=p)  First Success on Trial 1  S, y = 1  p(1)=p  First Success on Trial 2  FS, y = 2  p(2)=(1-p)p  First Success on Trial k  F…FS, y = k  p(k)=(1-p) k-1 p

5. Binomial Distribution – Expected Value

6. Geometric Distribution – Expected Value

7. Binomial Distribution – Variance and SD

8. Geometric Distribution – Variance and SD

9. Binomial Distribution for On-Time Flights

10. Binomial Distributions for n=1,2,3,4,10,25 In EXCEL:  Create a column of values 0,1,2,…,n (Say 0 is in cell A2)  In Cell B2, Type: =BINOM.DIST(A2,n,p,0)  Copy and paste that cell alongside 1 (A3),…,n  Note that the 0 at the end gives P(Y = y) = p(y)  If you use 1 instead, you get P(Y ≤ y) = F(y)

11. Several Binomial Distributions with p=0.8

18. Geometric Distribution Probabilities

19. Geometric Distribution In EXCEL:  Create a column of values 1,2,…,Y* for some large value of Y* (Say 1 is in cell A2)  In Cell B2, Type: =NEGBINOM.DIST(A2-1,1,p,0)  Copy and paste that cell alongside 1 (A3),…,Y*  Note that the 0 at the end gives P(Y = y) = p(y)  If you use 1 instead, you get P(Y ≤ y) = F(y)

20. Geometric Distribution Probabilities and CDF

21. Moment-Generating Function

22. Moment-Generating Function – Binomial Distribution

23. Geometric Distribution – MGF

24. Probability-Generating Functions

25. Probability-Generating Functions - Binomial

26. Geometric Distribution – PGF