Discrete Probability Distributions

1. Discrete Probability Distributions

2. Discrete vs. Continuous • Discrete • A random variable (RV) that can take only certain values along an interval: • Cars passing by a point • Results of coin toss • Students taking a class • Continuous • An RV that can take on any value at any point along an interval: • Temperature, time, distance, money etc.

3. Frequency Distribution • Number of times an observation occurs in a given population.

4. What is a variable? • A symbol (A, B, x, y, etc.) that can take on any of a specific set of values • X=number of heads • Y= temperature • Random variable • The outcome of a statistical experiment

5. Random variable notation • Capital letter represents the RV • X=total number of heads in 4 tosses • P(X) represents the probability of X • Lower-case letter represents one of the values of the RV • P(X=x) is the probability the RV will assume a specific value • P(X=2) is the probability that we will have exactly 2 heads in the 4 tosses

6. Probability Distribution • Relative frequency distribution that should, theoretically, occur for observations from a given population.

7. Cumulative probability distribution • Probability the value of a RV falls within a specified range. • Coin toss: P(X≤1)

8. Characteristics of a Discrete Probability Distribution • For any value of x • The values of x are exhaustive, i.e. the distribution contains all the possible values • The values of x are mutually exclusive; i.e., only one value can occur for an experiment • The sum of the probabilities equals 1

9. Mean and standard deviation • Mean of discrete distribution is called expected value • Variance • Standard Deviation

10. Practice • Determine the Mean (µ) or Expected Value (E(x)) for the following data.

11. Practice • A music shop is holding a promotion in which the customer rolls a die and deducts a dollar from the price of a CD equal to the number that he rolls. • If the owner pays $5.00 for each disk and prices them at $9.00, what will his expected profit be on each CD during this promotion?

12. Binomial Distributions • There are 2 or more identical trials • In each trial, there can be only 2 outcomes (success or failure) • Trials are statistically independent • Outcome of one trial does not influence outcome of the next • Probability of success remains the same from one trial to the next

13. Binomial Experiment? • An article in a 1988 issue of The New England Journal of Medicine talked about a TB outbreak. • One person caught the disease in 1995 • 232 workers sampled from a very large population were given a TB test • The number of workers testing positive is the variable of interest • If we test all 232 workers for the disease, is this a binomial experiment?

14. Binomial Experiment? • Bill has to sell 3 cars to meet his monthly quota. He has 5 customers, but 3 of them are interested in the same car and will leave if that car is sold. • He has a 30% chance of a sale with each customer. • Is this a binomial experiment?

15. Binomial Distributions • Probability of exactly x successes in n trials: • Where: • π = probability of success for any trial • n = number of trials • x = number of successes • (1-x) = number of failures

16. Binomial Distributions • Expected value • Variance

17. Binomial Distributions in Excel • =binom.dist(number_s,trials,probability_s,cumulative) • Where: • Number_s = number of successes • Trials • Probability_s = probability of success • Cumulative: • False, if we want the probability of x • True, if we want the probability of all the variables up to and including x • Example, in the previous problem • P(X=5) • =binom.dist(5,5,.1,false)

18. Binomial Experiment • We’re going to select 5 households at random in a city where the unemployment rate is 10% to see if the head of the household is unemployed. What is the probability that all 5 are employed? • Is this a binomial experiment? • Why or why not?

19. These examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=Stat These examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=Stat These examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=Stat Acceptance to college • The probability that a student is accepted to a prestigious college is 0.3. If 5 students apply, what is the probability that at most 2 are accepted?

20. Probability Distribution Probability # Accepted

21. Cumulative Probability Distribution Probability # Accepted

22. Coin flipping - again • What is the probability of getting 45 or fewer heads in 100 tosses of a fair coin?

23. Probability Distribution

24. Cumulative Probability Distribution

25. The World Series • What is the probability that the World Series will last 4 games? • 5 games? • 6 games? • 7 games? • Assume the teams are evenly matched.

26. Poisson Distribution • Applies for events occurring over time, space, or distance • Examples: • Number of cars driving past a point • Number of defects per foot in manufactured pipe • Number of knots in a section of wood panel • Number of accidents per day at a job site

27. Poisson Distribution e is the base of the natural logarithm system and is equal to 2.71828 Any number raised to a negative exponent is the same as 1 divided by that number raised to its exponent. Example: 2-2 is the same as 1/22

28. Poisson Distribution • There were 438 children born in a small town last year. • What is the probability that, on any given day, no children were born?

29. Poisson Distribution in Excel • =poisson.dist(x,mean,cumulative) • Where: • X=the number we’re looking for • Mean = lambda • Cumulative • True = probability of all values up to and including x • False = probability of x • =poisson.dist(0,1.2,false)

30. Probability Distribution

31. Cumulative Poisson Distribution • Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

32. Cumulative Poisson Distribution

33. Hypergeometric Distribution • Sampling without replacement • Compare to binomial • There are 2 or more identical trials • In each trial, there can be only 2 outcomes (success or failure) • Trials are statistically independent • Probability of success remains the same from one trial to the next • The random variable is the number of successes in n trials Trials are not statistically independent Probability of success changes from one trial to the next

34. Hypergeometric Distribution Where: N=size of the population n=size of the sample s=number of successes in the population x=number of successes in the sample

35. Hypergeometric in Excel • =hypgeom.dist(sample_s, number_sample, population_s, number_population, cumulative) • Where: sample_s=number of successes in the sample number_sample=size of the sample population_s=number of successes in the population number_population=size of the population cumulative=same as before • =hypgeom.dist(2, 4, 6, 20, false)

36. Hypergeometric Distribution • 20 businesses filed tax returns • 6 of the returns were filled out incorrectly • The IRS has randomly selected 4 of the 20 returns to audit • What is the probability that exactly 2 of the 4 selected for audit will be filled out incorrectly?

37. Hypergeometric Distribution

38. Cumulative Hypergeometric • Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts?

39. Cumulative Hypergeometric

40. Summary • Random variables • Discrete v. continuous • Probability distributions • Cumulative distributions • Expected values • Binomial • Poisson • Hypergeometric