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08 Discrete Probability Distributions Part 2

3-4 3 2.6*4*3)4.1(4.13221==-=)1(xp1515binomial

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Unformatted text preview: 3--4 3 2.6.*4.*3)4.1(4.13221==-=)1(XP1515Binomial distribution: Example1535% of all voters support Proposition A. If a random sample of 10 voters is polled, what is the probability that exactly three of them support the proposition?1616Binomial distribution: Example1635% of all voters support Proposition A. If a random sample of 10 voters is polled, what is the probability that exactly three of them support the proposition?i.e., find P(x= 3) if n= 10 and p= 0.35There is a 25.22% chance that 3 out of the 10 voters will support Proposition A..2522(0.65)(0.35)3!7!10!qpx)!(nx!n!3)P(x73xnx==-==-17Binomial distribution in Excel: X ~ Binomial (n, p)17=BINOMDIST( x , n, p, 0/FALSE)=BINOMDIST( x , n, p, 1/TRUE)X ~ Binomial (n = 10, p = 0.35)P(X = 3)P(X ≤ 3) (= P(X=0) + P(X=1) + P(X=2) + P(X=3))xnxppxnxXPxP--===)1()()(inixippinxFxXP-=-==≤∑)1()()(1818Binomial example: Alumni pledge solicitor18Sarah is working in the Drexel Alumni Association, calling up alum and soliciting funds for a new scholarship program. Suppose the probability that Sarah wins a pledge for funds from a randomly chosen target alum is 1/4. If she calls 8 alums this evening, what is the probability that:1.No money will be pledged?2.Exactly two pledges will be made?Binomial (n=8, p=0.25)P(X=2) = BINOMDIST(2,8,0.25,0) = 0.31151001.75.25.8)(8===XP1919Binomial example: Alumni pledge solicitor193.At least two pledges will be made?4.Sarah is paid $6.00 per hour for this job and makes 8 calls per hour. Suppose she gets a bonus of $5.00 for each successful call. What is her expected earning per hour?P(X≥2) = 1 – P(X≤1) = 1 – BINOMDIST(1,8,0.25,1)= 0.6329E[X] = np = 2E[bonus] = 5*E[X] = 10Hourly pay = 6Exp hourly earning = 1620In Binomial distribution, each trial is independent and identical. (Sampling with replacement)If we draw n candies one at a time (out of N total), depending on what candies are drawn already, the probability that a particular candy is drawn next changes. (Sampling without replacement)Does the number of successes out of n follow a binomial? (Yes or No). As the size of the total candy pool (N) increases, Sampling without replacement converges to sampling with replacementBinomial distribution: Caveat (Effect of finite population)21Discrete random variable: Poisson distribution Poisson distribution: Used to count the number of occurrences of a particular event during an interval of time or over a space For example: The number of customer arrivals in an hourThe number of calls received at a call center in a two-hour shiftThe number of defects on a 17 inch LCD screen22Poisson distribution Properties and assumptions of the poisson distributionCan take any non-negative integervalue (x=0,1,2,3,…….):the average number of outcomes of interest per unit time segment or unit space segment....
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3-4 3 2.6*4*3)4.1(4.13221==-=)1(XP1515Binomial distribution...

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