Discrete Random Variables and Probability Distributions

1. Discrete Random Variables and Probability Distributions Dr. Papia Sultana Associate Professor Department of Statistics Rajshahi University

2. Outline • Probability distribution • Probability distribution function and mass function • Parent distribution • Binomial distribution • Poisson distribution • Geometric distribution • Negative Binomial distribution • Hyper-geometric distribution • Multinomial distribution Dr. Papia Sultana

3. Probability distribution • Arrangement of probabilities of outcome of an event. • Ex. Tossing three coins simultaneously. Outcome of interest is “number of head” (X) in a toss. Values of X: 0 1 2 3 (No of Head) (TTT) (TT H) (T HH) (HHH) Probabilities: 1/8 3/8 3/8 1/8 Dr. Papia Sultana

4. Probability distribution function Let X be a random variable, then the function is called distribution function of X. Ex. Consider previous example. Values of X: 0 1 2 3 (x) F(x): 1/8 4/8 7/8 1 Dr. Papia Sultana

5. Distribution mass function Let X be a one-dimensional discrete random variable, then the probability function is called the probability mass function if it satisfies (a) (b) Dr. Papia Sultana

6. Parent distribution The distribution of a true sample. It is, actually, distribution of population. It can be obtained limiting the parameters to the true value of distribution of any sample. Binomial Uniform Poisson Normal Geometric Gamma Negative Binomial Weibul Hyper-geometric Cauchy Multinomial Beta .............so on Dr. Papia Sultana

7. Binomial distribution Bernoulli trials: Tossing a coin: Head or Tail More Example: Germination of a seed, gender of newborn baby, defectiveness of products of a factory, ........... Dr. Papia Sultana

8. Binomial distribution Let us consider n Bernoullian trials with outcome “success” or “failure”. SSFSFFFSFSSFF...........FSF # of S=X, # of F=n-X, P q=1-p Dr. Papia Sultana

9. Binomial distribution These X success out of n trial can occur in ways with probability . Dr. Papia Sultana

10. Binomial distribution Properties: • mean=np, variance=npq, skewness= , kurtosis= • mgf: • It does not follow additive properties. Dr. Papia Sultana

11. Binomialdistribution • Example: The number X of seeds that germinate in n=10 independent trials with p=0.8 is b(10,0.8), that is, • Distribution function Dr. Papia Sultana

12. Binomial distribution Dr. Papia Sultana

13. Poisson distribution • Number of customers coming at a restaurant per hour. • Number of patients coming to take service at a clinic per day. • Number of telephone calls at a helpline center per minute. • ......... Dr. Papia Sultana

14. Poisson distribution • Let the probability of success for each trial is . • Probability of success for each trial is indefinitely small. • n is sufficiently large (much larger than ). Dr. Papia Sultana

15. Poisson distribution pdf is Dr. Papia Sultana

16. Poisson distribution • Example: Telephone calls enter a Hall switchboard of Rajshahi University on the average two every 3 minutes. Therefore, per 3-minutes period and pdf is Dr. Papia Sultana

17. Poisson distribution Properties: • mean=variance= , skewness= , kurtosis= • mgf: • It follows additive properties. Dr. Papia Sultana

18. Poisson distribution Dr. Papia Sultana

19. Geometric distribution Dr. Papia Sultana

20. Geometric distribution Example: Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70 and the number of successes is 1 . During the season, what is the probability that Bob makes his first free throw on his fifth shot? The probability of success (P) is 0.70, the number of trials (x) before the success is 4 Dr. Papia Sultana

21. Geometric distribution Properties: • mean= ,variance= , • mgf: • It does not follow additive properties. Dr. Papia Sultana

22. Geometric distribution Properties: • mean= ,variance= , • mgf: • It does not follow additive properties. Dr. Papia Sultana

23. Negative-Binomial distribution When variance is larger than mean. Example: • Deaths of insects, • Insect bites, • ............. Dr. Papia Sultana

24. Negative-Binomial distribution • Last trial must be a success. • There are x failures preceding the r-th success. Dr. Papia Sultana

25. Negative-Binomial distribution Properties: • mean= ,variance= , • mgf: • It does not follow additive properties. Dr. Papia Sultana

26. Negative-Binomial distribution • Example: Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot? This is an example of a negative binomial experiment. The probability of success (P) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 3. Thus, the probability that Bob will make his third successful free throw on his fifth shot is 0.18522. Dr. Papia Sultana

27. Hypergeometric distribution • When the population is finite and the sampling is done without replacement, so that the events are stochastically dependent, although random. Dr. Papia Sultana

28. Hypergeometric distribution • Consider an urn with N balls, M of which are white and N-M are red. Suppose that we draw a sample of n balls at random (without replacement) from the urn, then the probability of getting k white balls out of n (k<n) will follow hypergeometric distribution. Dr. Papia Sultana

29. Hypergeometric distribution • pdf: Hg(N,M,n) Dr. Papia Sultana

30. Hypergeometric distribution • Applications: • Industrial acceptance: An acceptance sampling scheme is based upon drawing a random sample of size n from a batch of N items, assumed to contain an unknown number d of defectives. If the number of defectives X in the sample is too large, the batch as whole is rejected. Deciding on the threshold to determine what “too large” should mean is based upon the fact that X~Hg(N,n,d). The parameter of interest is d. Dr. Papia Sultana

31. Hypergeometric distribution • Capture-recapture experiment: A population of N individuals of a species exists in a closed eco-system. a first random sample of size is taken, all are tagged and returned to the wild. Later asecond random sample of size is taken. let X be the number of tagged individuals caught in the second sample. Then and the parameter of interest is N. Dr. Papia Sultana

32. Hypergeometric distribution • System faults: A system has an unknown number N of faults, and two independent inspection processes detect , faults respectively, of which X are common. Then and the parameter of interest is N. Dr. Papia Sultana

33. Hypergeometric distribution • Example: In a pond there were 200 fishes. A catch of 50 fishes made and returned them alive into the pond marking each with a red spot. After a reasonable period of time, another catch of 30 fishes was made. what is the probability that exactly 11 of the spotted fishes were caught? Dr. Papia Sultana

34. Hypergeometric distribution Dr. Papia Sultana

35. Hypergeometric distribution Properties: • mean= ,variance= , Dr. Papia Sultana

36. Multinomial distribution • This is an generalization of binomial distribution. • When there are more than two outcomes of a trial, it follows multinomial distribution. Dr. Papia Sultana

37. Multinomial distribution • Example: In a large population of plants, there are three possible alleles S, I and F at one locus resulting in six genotypes SS,II, FF, SI, SF and IF. Let denote the allele frequencies of S, I, F. The labels and frequencies of the genotypes are Label: 1 2 3 4 5 6 Genotype: SS II FF SI SF IF Frequency: HWE Freq: Dr. Papia Sultana

38. Multinomial distribution • Let are k mutually exclusive and exhaustive outcomes of a trial with probabilities . Let occurs times, occurs times, ......, and occurs times. Dr. Papia Sultana

39. Multinomial distribution Properties: • mgf: Dr. Papia Sultana

40. Multinomial distribution • Example: In a recent three-way election for a country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes. If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters for candidate B and three supporters for candidate C in the sample? Dr. Papia Sultana

41. Thank you! Dr. Papia Sultana