Hypergeometric & Poisson Distributions Dhon G. Dungca, M.Eng’g.

Hypergeometric Distribution Is the appropriate distribution when sampling without experiment is used in a situation that could qualify as a Bernoulli Process. The probability distribution of the hypergeometric distribution of the hypergeometric variable x is the number of successes in a random sample size n selected from N items of which k are classified as sucesses, and n-k as failures. Its variables are denoted by P(x; N, n, k).

Hypergeometric Distribution The formula for determining the probability of a designated number of successes x is: k N-k h(x; N, n, k) = x n-x N n

Hypergeometric Distribution Where: N – total # of items in population k – number of successes N-k – number of failures n – total # of items in the sample x – number of successes n-x – number of failures

Example 1 If 7 cards are drawn from a deck of playing cards, what is the probability that two will be spades? 0.3357

Example 2 Lots of 40 components each are called unacceptable if they contain as many as 3 defectives or more. The procedure for sampling the lot is to select 5 components at random and to reject a lot if a defective is found. What is the probability that exactly 1 defective is found if there are 3 defectives in the entire lot? 0.3011

Example 3 A homeowner plants 6 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 2 daffodil bulbs and 4 tulip bulbs? 5/14

Example 4 To avoid detection at customs, a traveller has placed 6 narcotic tablets in a bottle containing 9 vitamin pills that are similar in appearance. If a customs official selects 3 tablets at random for analysis, what is the probability that the traveller is arrested for illegal possession of narcotics? 0.8154

Example 5 A company is interested in evaluating its current inspection procedure on shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to be defective. What proportion of 20% defective shipments will be accepted? 0.9517

Poisson Distribution Experiments yielding numerical values of a random variable X, the number of outcomes occurring during a given time interval or in a specified region, are called Poisson Experiments The given time interval may be of any length, such as a minutes, a day, a week, a month or even a year.

Poisson Distribution The probability distribution of the Poisson random variable X, representing the number of outcomes occurring in a given time interval or specified region denoted by t, is: p(x; ) = e - ()x x!

Poisson Distribution Where: e = 2.71828… = is the average number of successes occurring per stated unit. x = the number of occurrences of the event in the interval of size t.

Example 1 If the number of complaints received daily by the Complaint department of a big department store in Makati is a random variable having a Poisson Distribution mean  = 5, what is the probability that exactly two complaints will be receive on any randomly selected day? 0.0842

Example 2 In a research study, the average train arrival per hour is 1. What is the probability of exactly 2 arrivals in a given hour? What is the probability of having 2 or more arrivals in 1 hour? What is the probability that there will be between 2 and 5 arrivals, inclusive in 1 hour? What is the probability that there will be 1 arrival between 2pm and 230pm? 0.1839, 0.2642, 0.2636, 0.3033

Example 3 The average quantity demanded for a certain signature item is 5 per day. What is the probability that there will be 20 of these items demanded in a week if a week consists of 6 working days. 0.0134

Example 4 Service calls come to a maintenance center according to a Poisson process and on the average 2.7 calls come per minute. Find the probability that No more than 4 calls come in any minute Fewer than 2 calls come in any minute More than 10 calls come in a 5-minute period 0.8629, 0.24866, 0.7888

Example 5 On the average, a certain intersection results in 3 traffic accidents per month. What is the probability that in any given month at this intersection Exactly 5 accidents will occur? Less than 3 accidents will occur? At least 2 accidents will occur? 0.1008, 0.4232, 0.8009