Discrete Probability Distributions Binomial Distribution Poisson Distribution Hypergeometric Distribution

Binomial Probability Formula

Binomial Probability Distribution By listing the possible values of x with the corresponding probability of each, we can construct a Binomial Probability Distribution.

Constructing a Binomial Distribution In a survey, a company asked their workers and retirees to name their expected sources of retirement income. Seven workers who participated in the survey were asked whether they expect to rely on Pension for retirement income. 36% of the workers responded that they rely on Pension only. Create a binomial probability distribution.

Constructing a Binomial Distribution x P(x) 0.044 1 0.173 2 0.292 3 0.274 4 0.154 5 0.052 6 0.010 7 0.001 P(x) = 1 Notice all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.

Population Parameters of a Binomial Distribution Mean:  = np Variance: 2 = npq Standard Deviation:  = √npq

Example In Murree, 57% of the days in a year are cloudy. Find the mean, variance, and standard deviation for the number of cloudy days during the month of June. Mean:  = np = 30(0.57) = 17.1 Variance: 2 = npq = 30(0.57)(0.43) = 7.353 Standard Deviation:  = √npq = √7.353 ≈2.71

Problem 1 Four fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of Heads and compute the probabilities of obtaining: No Heads Precisely 1 Head At least 1 Head Not more than 3 Heads

Problem 2 If the Probability of hitting a target in a single shot is 10% and 10 shots are fired independently. What is the probability that the target will be hit at least once?

Poisson Process The Poisson Process is a counting that counts the number of occurrences of some specific event through time. Number of customers arriving to a counter Number of calls received at a telephone exchange Number of packets entering a queue

Poisson Probability Distribution The Poisson probability distribution provides a good model for the probability distribution of the number of ‘rare events’ that occur randomly in time, distance, or space.

Assumptions Poisson Probability Distribution The probability of an occurrence of an event is constant for all subintervals and independent events There is no known limit on the number on successes during the interval As the unit gets smaller, the probability that two or more events will occur approaches zero.

µ = 1 µ = 4 µ = 10

Poisson Probability Distribution f(x) = The probability of x successes over a given period of time or space, given µ µ = The expected number of successes per time or space unit; µ > 0 e = 2.71828 (the base for natural logarithms)

Problem 5 Let X be the number of cars per minute passing a certain point of some road between 8 A.M and 10 A.M on a Sunday. Assume that X has a Poisson distribution with mean 5. Find the probability of observing 3 or fewer cars during any given minute.

Problem 7 In 1910, E. Rutherford and H. Geiger showed experimentally that number of alpha particles emitted per second in a radioactive process is random variable X having a Poisson distribution. If X has mean 0.5. What is the probability of observing 2 or more particles during any given second?

Problem 9 Suppose that in the production of 50 Ω resistors, non-defective items are those that have a resistance between 45 Ω and 55 Ω and the probability of being defective is 0.2%. The resistors are sold in a lot of 100, with the guarantee that all resistors are non-defective. What is the probability that a given lot will violate this guarantee?

Problem 11 Let P = 1% be the probability that a certain type of light bulb will fail in 24 hours test. Find the probability that a sign consisting of 100 such bulbs will burn 24 hours with no bulb failures.

Multinomial Distribution If a given trial can result in K outcomes E1,E2, …, Ek with probabilities p1,p2, …,pk, then the Probability Distribution of the random variables X1,X2, …, Xk, representing the number of occurrences for E1,E2, …, Ek in n independent trials is

Example An airport has three runways. The probabilities that the individual runways are accessed by a randomly arriving commercial jets are as following: Runway 1: p1 = 2/9 Runway 2: p1 = 1/6 Runway 3: p1 = 11/18 What is the probability that 6 randomly arriving airplanes are distributed in the following fashion? Runway 1: 2 airplanes Runway 2: 1 airplanes Runway 3: 3 airplanes

Sampling With Replacement

Hypergeometric Probability Distribution In cases where the sample size is relatively large compared to the population, a discrete distribution called hypergeometric may be useful.

Sampling Without Replacement Hypergeometric Distribution = Different ways of picking n things from N = Different ways of picking x defective from M = Different ways of picking n-x nondefective from N-M

Hypergeometric Distribution Mean and Variance

Problem 13 Suppose that a test for extra sensory perception consists of naming (in any order) 3 cards randomly drawn from a deck of 13 cards. Find the probability that by chance alone, the person will correctly name (a) no cards, (b) 1 Card, (c) 2 Cards, and (d) 3 cards.

Quiz # 2 32 Cptr (B) – 5 NOV 2012 If the Probability of hitting a target in a single shot is 5% and 20 shots are fired independently. What is the probability that the target will be hit at least once?