08 Discrete Probability Distributions Part 2

3 432 6 4 3 4 1 4 1 3 2 2 1 1 x p 15 binomial

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3 - - 432 . 0 6 . 0 * 4 . 0 * 3 ) 4 . 0 1 ( 4 . 0 1 3 2 2 1 = = - = ) 1 ( X P
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15 Binomial distribution: Example 35% 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?
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16 Binomial distribution: Example 35% 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 = 0.35 There is a 25.22% chance that 3 out of the 10 voters will support Proposition A. .2522 0 (0.65) (0.35) 3!7! 10! q p x)! (n x! n! 3) P(x 7 3 x n x = = - = = -
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17 Binomial 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)) x n x p p x n x X P x P - - = = = ) 1 ( ) ( ) ( i n i x i p p i n x F x X P - = - = = ) 1 ( ) ( ) ( 0
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18 Binomial example: Alumni pledge solicitor Sarah 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) = 0.3115 1001 . 0 75 . 0 25 . 0 0 8 ) 0 ( 8 0 = = = X P
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19 Binomial example: Alumni pledge solicitor 3. 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) = 0.6329 E[X] = np = 2 E[bonus] = 5*E[X] = 10 Hourly pay = 6 Exp hourly earning = 16
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20 In 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 replacement Binomial distribution: Caveat (Effect of finite population)
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21 Discrete 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 hour The number of calls received at a call center in a two- hour shift The number of defects on a 17 inch LCD screen
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