Poisson Distribution The number of defects in one
Poisson Distribution The number of defects in one mile of 405 freeway. The number of vehicles arriving at Costco gas station on Tampa Avenue in 30 -minute intervals. The number of air planes arriving LAX in one hour intervals. The number of emails sent out in DNCBE in one minute intervals. The link to the excel file http: //www. csun. edu/~aa 2035/Course. Base/Probability/S-3 b-Binomial/Poisson. xlsx
Poisson Probability Distribution Poisson Distribution 1 p p p Poisson Distribution 2 A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space. It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . . ). Two properties of Poisson Distribution n n The probability of an occurrence is the same for any two intervals of equal length. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence other intervals. Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 2
Poisson Probability Distribution Formula x = the number of occurrences in an interval f(x) = the probability of x occurrences in an interval µ = mean number of occurrences in an interval e = 2. 71828 x! = x(x – 1)(x – 2). . . (2)(1) p p Theoretically, there is no upper limit for the number of occurrences, therefore, probability function f(x) is defined for all positive values x = 0, 1, 2, … with no limit. In practical, f(x) after an initial increase, starts decreasing, such that after some points it is approximately zero. Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 3
Poisson Probabilities Using Math Formulas Every 10 minutes 8 airplanes arrive at LAX. m = 8 arrivals per 10 min. Radom Variable x shows the number of arrivals in 10 mins. Use Poison formula to compute P(x) = 1, 2, 3, …. 10. Also compute P(x) ≤ 1, 2, 3, …. 10. Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 4
Poisson Probabilities Using Excel Formulas Solve the same problem using the Poisson Distribution formula in excel. P(x) = 1, 2, 3, …. 20, also compute P(x) ≤ 1, 2, 3, …. 20. Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 5
Examples of Poisson Probability Distribution The number of telephone calls that pass through a switchboard has a Poisson distribution with mean equal to 2 per minute. The expected number of phone calls that pass through the switchboard in one minute is a) 2 b) 4 c) 3 d) 1 2 In a certain communication system, there is an average of 1 transmission error per 10 seconds. Let the distribution of transmission errors be Poisson. What is the probability of having 2 errors in one-half minute in duration? a. 0. 184 b. 0. 224 c. 0. 448 d. 0. 0498 e. 0. 378 Mean = 3(1) = 3 POISSON. DIST(2, 3, 0) = 0. 22404 Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 6
Examples of Poisson Probability Distribution The number of customers entering a bank per minute is a Poisson random variable with a mean of 3. 5 customers per minute. What is the probability that more than three customers enter the bank in a minute? a. 0. 3209 b. 0. 4633 c. 0. 5367 d. 0. 6791 P(x≥ 3) = 1 - P(x≤ 2) 1 - POISSON. DIST(2, 3. 5, 1) 1 - 0. 32085 = 0. 67915 The marketing manager of a company usually receives 10 complaint calls during a week (consisting of five working days). Suppose that the number of calls during a week follows the Poisson distribution. The probability that she gets five such calls in one day is: a. 0. 0361 b. 0. 0378 c. 0. 9834 d. 0. 2000 P(x= 5) = POISSON. DIST(5, 10, 0) = 0. 03783 Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 7
Examples of Poisson Probability Distribution A retail store aimed to reduce the number of bad checks cashed by its cashiers. The store's goal is to cash no more than eight bad checks per week. The average number of bad checks cashed is three per week. a) Find the probability that the store's cashiers will not cash any bad checks in a particular week. P(x=0) = POISSON. DIST(0, 3, 0) = 0. 04979 = b. Find the probability that the store will not meet its goal in a particular week. P(x>8) = 1 - P(x≤ 8) = 1 - POISSON. DIST(8, 3, 1) = 1 - 0. 9962 = 0. 0038 Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 8
Examples of Poisson Probability Distribution c. Find the probability that the store's cashiers will cash no more than 4 bad checks per two-week period. Mean for 2 weeks = 2(3) = 6 P(x≤ 4) =POISSON. DIST(4, 6, 1) = 0. 28506 d. Find the mean, variance, and standard deviation of the number of bad checks per week? 3, 3, SQRT(3) Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 9
Examples of Poisson Probability Distribution m = s 2 = 10 s = SQRT(10 Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 10
Poisson Graph, Probabilities Graph Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 11
Poisson and Binomial Suppose 60% of a large group of animals is infected with a particular disease. Let Y= the number of non- infected animals in a sample of size 5. The distribution of Y is n a) binomial with n = 5 and p = 0. 6 n b) binomial with n = 5 and p = 0. 4 n c) binomial with n = 5 and p = 0. 5 n d) Poisson with λ =0. 6 Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 12
Poisson and Binomial A typical page in a book contains one typo per page. What is the probability that there are exactly 8 typos in a given 10 -page chapter? This is a Poisson distribution. Since the expected number of typos on one page is 1, the expected number of typos in 10 pages is 10. λ = 10, P(x=8) =? Suppose that every page in the chapter contains exactly 500 words , and there is still an average of one typo per page. What is the probability that there are exactly 8 typos in the 10 -page chapter? This time, we use a binomial distribution, where a typo counts as a success. The probability of success is 1/500 ; we have 5000 trials. Descriptive Statistics Ardavan Asef-Vaziri Jan-2018 13
- Slides: 13