Sec3.4-3.6_427

Sec3.4-3.6_427 - Sect 3.4 - 3. 6: Special, useful RVs....

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1 Sect 3.4 - 3. 6: Special, useful RV’s. Certain distributions very often arise in practice Organize presentation and study around: 1. Recognition: Definition, settings and uses, assumptions. 2. Prob Dist; pmf 3. Properties: mean, variance, etc. 4. Computations: Mainly, how to find probabilities.
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2 Example: Uniform Discrete uniform RV X takes on one of k values, x 1 , x 2 , …, x k with equal probabilities. p(x) = 1/k, for x = x 1 , x 2 , …, x k and 0 else. Language: “uniform dist” means prob dist is constant
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3 Example: Bernoulli (p. 88) 1. A Bernoulli RV X takes on values 1 or 0. Assumption: P(X = 1) = p . 2. p(x ; p) = p x (1- p) 1-x , for x = 0, 1; 0 else. 3. m = 1 (p ) + 0 (1 - p ) = p s 2 = E(X 2 ) – m 2 = p (1 - p)
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4 Sect 3.4 Binomial RV 1. Experiment consists of n repeated trials 2. There are only two possible outcomes on each trial. Ex) male or female, defect or not defect, success (S) or failure (F), 1 or 0, etc. 3. Trials are independent 4. Prob. of success is p on each trial. Then X = the number of successes is a binomial RV. Shorthand: X ~ Bin(n, p). Read ~ as “is distributed as a”
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5 pmf for x = 0, 1, …, n; 0 else Ex) Roll a fair die 3 times. Let X = number of “4”s in n=3 rolls. X is binomial, with “4” a Success (S), n = 3, P(S) = p = 1/6 . Find P( X = 2). x n x p p x n p n x b x p ) 1 ( ) , ; ( ) (
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6 ( n=3, P(S) = p = 1/6 ) P(X =2) = P(S,S,F) + P(S,F,S) + P(F,S,S) (“prob. event =sum of prob.’s of the ways it can occur”) = P(S)P(S)P(F) + P(S)P(F)P(S) + P(F)P(S)P(S) (independence) (1/6) 2 (5/6) 3-2 p x (1- p) n-x x n 2 3 # of ways for x S’s in n trials Prob of each way
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7 Properties n=1: binomial is Bernoulli If X ~ Bin(n, p) and Y = number of failures, then Y ~ Bin(n, 1- p) mean m = np variance s 2 = np(1- p)
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8 X ~ Bin(n,p), show E(X) = np For x > 1 so (set y=x 1) 1 1 )! ( )! 1 ( )! 1 ( )! ( )! 1 ( ! )! ( ! ! x n n x n x n n x n x n x n x n x np p p y n np p p x n np p p x n n p p x n x X E y n y n y x n x n x x n x n x x n x n x 1 1 0 1 1 1 0 ) 1 ( 1 ) 1 ( 1 1 ) 1 ( 1 1 ) 1 ( ) (
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Finding Prob.s: small n: tables, computer/calc. moderate n: computer/calc.
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Sec3.4-3.6_427 - Sect 3.4 - 3. 6: Special, useful RVs....

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