1. Discrete Probability Distributions Chapter 5 MSIS 111 Prof. Nick Dedeke

2. Learning Objectives Distinguish between discrete random variables and continuous random variables. Know how to determine the mean and variance of a discrete distribution. Identify the type of statistical experiments that can be described by the binomial distribution, and know how to work such problems.

3. Learning Objectives -- Continued Decide when to use the Poisson distribution in analyzing statistical experiments, and know how to work such problems. Decide when binomial distribution problems can be approximated by the Poisson distribution, and know how to work such problems.

4. Random Variable Random Variable -- a variable which associates a numerical value to the outcome of a chance experiment

5. Discrete vs. Continuous Distributions Discrete Random Variable – arise from counting experiments. It has a finite number of possible values or an infinite number of possible values that can be arranged in sequence Number of new subscribers to a magazine Number of bad checks received by a restaurant Number of absent employees on a given day Continuous Random Variable – arise from measuring experiments. It takes on values at every point over a given interval Current Ratio of a motorcycle distributorship Elapsed time between arrivals of bank customers Percent of the labor force that is unemployed

6. Discrete and Continuous Distributions Discrete Binomial distributions Poisson distributions Continuous Uniform distributions normal distributions exponential distributions T distributions chi-square distributions F distributions

7. Experiment Experiment: We want to flip a coin twice. What are all the possibilities of the experiment?

8. Experiment Experiment: We want to flip a coin twice. What are all the possibilities of the experiment? Outcomes possible: HH, HT, TH, TT To assign numerical variables to the outcomes, we have to define the random variables Let X be number of heads in outcome Let Y be number of tail in outcome

9. Deriving Random Variables for Experiment Experiment: We want to flip a coin twice. What are all the possibilities of the experiment? Outcomes possible: HH, HT, TH, TT Let X be number of heads in outcome Let Y be number of tails in outcome Outcomevalues of Xvalues of Y HH 2 0 HT 1 1 TH 1 1 TT 0 2

10. Exercise: Deriving Random Variables for Experiment Experiment: We want to flip a coin thrice. What are all the possibilities of the experiment? What are the random variables? Outcomes possible: Let X be number of heads in outcome Let Y be number of tails in outcome

11. Exercise: Deriving Discrete Probability Distribution Experiment: We flip a coin thrice. What is the probability distribution for X random variable the experiment? Let X be the number of heads in outcome. What is the probability distribution for Y random variable? Outcome values of X values of Y HHH30 HHT21 HTH21 HTT12 THH21 THT12 TTH12 TTT03 XFi P(x) 010.125 130.375 230.375 310.125 8 1.00

12. Exercise: Deriving Discrete Probability Distribution Experiment: We investigate the number of people in a gym that injured themselves before we registered. The data is provided below. What is the probability distribution for X random variable the experiment? Let X be the number of muscle injuries in outcome. Let X be the number of bone injuries in outcome. What is the probability distribution for Y random variable? Outcome values of X values of Y 1 st pool40 2 nd pool 21 3 rd pool 41 4 th pool 42 5 th pool 31 6 th pool 32 7 th pool 12 8 th pool 00

13. Example: Discrete Distributions & Graphs 012345012345 0.37 0.31 0.18 0.09 0.04 0.01 Number of Crises Probability Distribution of Daily Crises 0 0.1 0.2 0.3 0.4 0.5 012345 ProbabilityProbability Number of Crises

14. Requirements for a Discrete Probability Function Probabilities are between 0 and 1, inclusively Total of all probabilities equals 1

15. Requirements for a Discrete Probability Function -- Examples XP(X) 0 1 2 3.1.2.4.2.1 1.0 XP(X) 0 1 2 3 -.1.3.4.3.1 1.0 XP(X) 0 1 2 3.1.3.4.3.1 1.2 PROBABILITY DISTRIBUTION : YESNONO

16. Example: Mean of a Discrete Distribution X 0 1 2 3 P(X).1.2.4.2.1 -.1.0.4.3 1.0 XPX  ()  = 1.0

17. Variance and Standard Deviation of a Discrete Distribution X 0 1 2 3 P(X).1.2.4.2.1 -2 0 1 2 X  4101441014.4.2.0.2.4 1.2

18. Example: Mean of the Crises Data XP(X) X  0.37.00 1.31 2.18.36 3.09.27 4.04.16 5.01.05 1.15 0 0.1 0.2 0.3 0.4 0.5 012345 ProbabilityProbability Number of Crises

19. Example: Variance and Standard Deviation of Crises Data XP(X) (X-  )  ) 2  ) 2  P(X) 0.37-1.151.32.49 1.31-0.150.02.01 2.180.850.72.13 3.091.853.42.31 4.042.858.12.32 5.013.8514.82.15 1.41

20. Exercise: Deriving Discrete Probability Distribution Experiment: We investigate the number of people in two gyms that injured themselves. The data is provided below. What is the probability distribution for X random variable for each experiment? What is the mean and standard deviations for each gym? What is the probability that equal to or greater than 3 injuries occur in each gym? Let X be the number of injuries in outcome. X1P(x1) 4 2 4 3 1 0 X2 P(x2) 4 2 3 4 3 2 1

21. Exercise: Deriving Discrete Probability For the data below, find the following probabilities: What is the probability that greater than 3 injuries occur in the gym? What is the probability that less than 2 injuries occur in the gym? What is the probability that greater than or equal to 2 but less than 4 injuries occur in the gym? Let X be the number of injuries in outcome. X1Fi P(x1) X*P(x1) 010.1250-2.6256.8906 0.0000 110.1250.125-1.6252.6406 2.6406 210.1250.250-0.6250.3906 0.7812 320.2500.7500.3750.1406 0.4218 430.3751.501.3751.8906 7.5624 8 1.002.625 11.953 11.406

22. Examples