08 Discrete Probability Distributions Part 2

8 binomial distribution examples a coin is flipped 10

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8 Binomial distribution: Examples A coin is flipped 10 times Heads or Tails X = number of heads. 10 parts are randomly sampled from a shipment of 30 parts Defective or Not X = number of nonconforming parts. 10 movies are randomly selected Profitable or Not X = number of profitable movies Investor buys options on 20 firms In the money or Not X = number “in the money” at exercise date
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9 Binomial distribution In the Binomial distribution, we are interested in X: the number of successes in the sample of size n (trials) where a success occurs with probability p on each trial. Xi= 1 if the i-th trial is a success. (with prob p) 0 otherwise. (with prob 1-p) X = the number of successes in the sample of n trials = X1 + X2 + … + Xn ) ( ) ( 3 2 1 x X X X X P x X P n = + + + = =
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10 Counting Rule for Combinations A combination is an outcome of an experiment where x objects are selected from a group of n objects where Cxn : number of combinations of x objects selected from n objects n! : n(n - 1)(n - 2) . . . (2)(1) x! : x(x - 1)(x - 2) . . . (2)(1) Note: 0! = 1 (by definition) C x n = n x ÷= n ! x !( n - x )!
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11 Binomial Distribution Formula P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, ( x = 0, 1, 2, ..., n ) n = random sample size p = probability of a success q = probability of a failure ( q = 1- p ) n ! = n ( n -1)( n -2)……1 0!= 1 (by definition) Probability mass function: P ( x ) = P ( X = x ) = n x ÷ p x q n - x = n ! x !( n - x )! p x q n - x
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S (p) S (p) S (p) S (p) S (p) S (p) S (p) F (1-p) F (1-p) F (1-p) F (1-p) F (1-p) F (1-p) F (1-p) (S,S,S) (p*p*p) (S,S,F) (p*p*(1-p)) (S,F,S) (p*(1-p)*p) (S,F,F) (p*(1-p)*(1-p)) (F,S,S) ((1-p)*p*p) (F,S,F) ((1-p)*p*(1-p)) (F,F,S) ((1-p)*(1-p)*p) (F,F,F) ((1-p)*(1-p)*(1-p)) Trial 1 Trial 2 Trial 3 This gives the number of ways to get x successes out of n trials This gives the probability of each of the ways of getting x successes in n trials x n x x n x p p x n x n p p x n x X P x P - - - - = - = = = ) 1 ( )! ( ! ! ) 1 ( ) ( ) (
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13 Binomial distribution Formula Mean: n = random sample size p = probability of a success n! = n(n-1)(n-2)……1 0!= 1(by definition) Variance and Standard Deviation: X = E[X] = 2= Var[X] = np npq σ = npq
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14 Binomial examples: X~ Binomial (n=3, p=0.4) Mean: X =E[ X ]= np Variance: 2=Var[ X ]= np(1-p) Standard deviation: = 1 * 1 * (0.6)3 = 0.216 P(X=0) + P(X=1) = 0.648 = 3 * 0.4 = 1.2 = 3 * 0.4 * 0.6 = 0.72 = √0.72 = 0.8485 1 – P(X ≤ 1) = 1 – 0.648 = = = ) 0 ( ) 0 ( X P P = = = ) 1 ( ) 1 ( X P P = = ) 1 ( ) 1 ( F X P 0 3 0 ) 4 . 0 1 ( 4 . 0 0
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