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Three_distributions - Prof M.Vaisfeld THE BINOMIAL...

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Prof. M.Vaisfeld THE BINOMIAL PROBABILITY DISTRIBUTION (see handout #6a) 6/8/2008 of the same basic experimental event. Each trial has only two possible outcomes independent trials is (#) x = 3 P(x) = 0.3125 mean µ = np = 2.5 n = 5 P(X ≤ x) = 0.8125 1.25 p = 0.5 1 – P(X ≤ x) = 0.1875 standard deviation σ = 1.118034 Example of Linear Histogram of a binomial probability distribution. 10 0.5 x P(X ≤ x) 0 1 0.0009765625 0.0009765625 1 10 0.009765625 0.0107421875 2 45 0.0439453125 0.0546875 3 120 0.1171875 0.171875 4 210 0.205078125 0.376953125 5 252 0.24609375 0.623046875 6 210 0.205078125 0.828125 7 120 0.1171875 0.9453125 8 45 0.0439453125 0.9892578125 9 10 0.009765625 0.9990234375 10 1 0.0009765625 1 obtained by means of MS Excel function =COMBIN(n, x) . BINOMIAL vs NORMAL DISTRIBUTION in the cells B51 and D51 (which display the copies of values stored in B29 and D29). 10 0.5 x f(x) = NORMDIST() 5 0 1 0.0009765625 0.0017000733 1.5811388 1 10 0.009765625 0.0102848443 2 45 0.0439453125 0.0417071001 3 120 0.1171875 0.1133716522 4 210 0.205078125 0.206576619 5 252 0.24609375 0.2523132522 6 210 0.205078125 0.206576619 7 120 0.1171875 0.1133716522 8 45 0.0439453125 0.0417071001 9 10 0.009765625 0.0102848443 10 1 0.0009765625 0.0017000733 Binomial Probability Experiment is an experiment that is made up of repeated trials (usually labeled as a success or a failure). There are n repeated independent trials, so that the probability of success, p , for each trial remains the same. The number x of successful trials (the binomial random variable) may take on any integer value from 0 to n . The binomial probability P(x) of exactly x successes and (n – x) failures in n P(x) = n C x p x (1 - p) n–x MS Excel has the function =BINOMDIST(x, n, p, cumulative ) where the variables x, n, p have the sense explained above, and cumulative is a logical value ( TRUE or FALSE ) that determines the form of the function. The function BINOMDIST(x, n, p, FALSE ) returns the probability P(x) (#) ; the function BINOMDIST(x, n, p, TRUE ) returns the cumulative distribution function P(X ≤ x) . To evaluate the probabilities P(x) , P(X ≤ x) , the mean µ , and the variance σ 2 type in the values of x, n , and p in the cells B23, B24 , and B25 respectively. variance σ 2 = np(1 – p) = n = Probability p = <== Vary the value of 0 ≤ p ≤ 1 in the cell D28 n C x P(x) = n C x p x (1 – p ) n–x The values of combinations n C x in the 2 nd column are Vary the values of n and p in the cells B29 and D29 respectively, not below n = Probability p = n C x P(x) = n C x p x (1 – p ) n–x µ = n* p = σ = [n p (1 - p )] 1/2 = A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 [some books recommend using this approximation only if np and
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