Chapter 4. Random Variables - 3

Outline Bernoulli and Binomial random variables Properties of Binomial Random Variables Poisson Random Variables Geometric Random Variables

Properties of Binomial RV – Expected Value

Properties of Binomial RV – Variance

Binomial Distribution

Proposition 6.1 If X is a binomial random variable with parameters (n, p), where 0<p<1, then as k goes from 0 to n, P{X=k} first increases monotonically and then decreases monotonically, reaching its largest value when k is the largest integer less than or equal to (n+1)p.

Computing Binomial Distribution Function

Poisson Random Variable - Examples Studying processes that generate rare, discrete number of events The number of wrong telephone numbers that are dialed in a day The number of customers entering a post office on a given day The number of vacancies occurring during a year in the federal judicial system The number of misprints on a page of a book The number of bacteria in a particular plate The number of people in a community living to 100 years of age

Poisson Random Variable The probability of i events in a time period t for a Poisson random variable with parameter  ( is also commonly used) is Parameter r represents expected number of events per unit time.  is the expected number of events over time period t. Difference between Binomial and Poisson distribution There are a finite number of trials n in Binomial distribution The number of events can be infinite for Poisson distribution

Probability mass function of Poisson RV

Poisson Distribution – Expected Value

Poisson Distribution – Variance

Ex 7a. Suppose that the number of typographical errors on a single page of a book has a Poisson distribution with parameter λ=1/2. Calculate the probability that there is at least one error on one page. X: the number of errors on this page P(X ≥ 1) = 1 - P(X = 0) P(X ≥ 1) = 1 - P(X = 0) = 1- e-1/2((1/2)0/0!) = 1 - e-1/2

Ex 7b. Suppose that the probability that an item produced by a certain machine will be defective is .1. Find the probability that a sample of 10 items will contain at most 1 defective item.

Ex 7c. Consider an experiment that consists of counting the number of α-particles given off in a 1-second interval by 1 gram of radioactive material. If we know from past experience that, on the average, 3.2 such α-particle are given off, what is a good approximation to the probability that no more than 2 α-particles will appear? The number of α-particles given off can be modeled by a Poisson random variable with parameter λ = 3.2.

Ex. If the area of a plate, A, is 100cm2 and there are r = Ex. If the area of a plate, A, is 100cm2 and there are r =.02 colonies per cm2, calculate the probability of at least 2 bacterial colonies on this plate. We have  = rA = 100(0.2) = 2. Let X = number of colonies.

Ex. The number of deaths attributable to polio during the years 1968-1976 is given in the following table. Based on this data set, can we use Poisson distribution to model the number of deaths from polio? ----------------------------------------------------------------------------- Year (19-) 68 69 70 71 72 73 74 75 76 --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- # deaths 24 13 7 18 2 10 3 9 16 The sample mean is 11.3 and the variance is 51.5. The Poisson distribution will not fit here because the mean and variance are too different.

Poisson Distribution – Weakly Correlated Events Poisson distribution remains a good approximation even when the trials are not independent, provided that their dependence is weak. Hat-matching problem, where n men randomly select hats from a set consisting of one hat from each person. What is the probability that there are k matches? We say that trial i is a success if person i selects his own hat. Ei = {trial i is a success} P(Ei) = 1/n The number of successes will approximately have a Poisson distribution with parameter n×1/n=1. The probability that there are k matches is e-1/k!. This is the same as obtained in Chapter 2, Ex 5m.

Poisson Random Variable as An Approximation for Binomial Random Variable Poisson random variable can be used as an approximation for a binomial random variable with parameters (n, p) when n is large and p is small so that np is a moderate size. Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10. λ = np.

Computing Poisson Distribution Function

Geometric Random Variable Independent trials, each having a probability p of being a success, are performed until a success occurs. If X is the number of trials required, X is a geometric variable with parameter p.

Ex 8a. An urn contains N white and M black balls Ex 8a. An urn contains N white and M black balls. Balls are randomly selected, one at a time, until a black one is obtained. If we assume that each selected ball is replaced before the next one is drawn, what is the probability that (a) exactly n draws are needed; (b) at least k draws are needed? If we let X denote the number of draws needed to select a black ball, then X is a geometric random variable with parameter M/(M+N).

Ex 8b. Find the expected value of a geometric random variable.

Find the variance of a geometric random variable. Compute E[X2] first.

Summary of Chapter 4 Random Variable Distribution Function Probability Density Function (Probability Mass Function) Expected value, mean, or expectation Variance Standard Deviation

Summary of Chapter 4 cont. Binomial Distribution Poisson Distribution Geometric Distribution