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
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Properties of Binomial RV – Variance
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Binomial Distribution
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
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Probability mass function of Poisson RV
Poisson Distribution – Expected Value
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• 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 pageP(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
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• 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.
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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.
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• 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.
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• 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.
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.
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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).
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Ex 8b. Find the expected value of a geometric random variable.
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