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l14_binombernoulli

l14_binombernoulli - Bernoulli Binomial Distributions...

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03/30/2009 CS206 - Intro. to Discrete Structures II 1 Bernoulli & Binomial Distributions Reading: Ross, Ch 4., Sec. 6. Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 2 Independent Trials If a random experiment is repeated several times under identical conditions, we perform independent trials of that experiment. E.g., Flipping a coin n times (under same conditions) Rolling two dice n times (under same conditions) In independent trials outcomes of any particular experiment do not depend on any future or past experiment outcomes. Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 3 Independent Trials - Definition Let E be an experiment on probability space ( Ω , P), E 2 {e 1 , …,e n }. Then, E n is the experiment of performing E n-times under identical conditions. E k is the k-th instance of experiment E. Ω n = { ω n =( ω 1 , ω 2 ,…, ω n ) | ω k 2 Ω , k=1,2,…,n and P( E 1 =i 1 ,E 2 =i 2 ,…,E n =i n ) = P(E 1 =i 1 ) ¢ P(E 2 =i 2 ) ¢ ¢ P(E n =i n ) = k=1 n P(E k =i k ) independence Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 4 Example Consider experiment E with three outcomes: Then, the sequence of two independent trials E 2 has the pmf E P(E=i) a 1/2 b 1/3 c 1/6 E 2 P(E 2 =(i 1 ,i 2 )) (a,a) ½ ¢ ½ = ¼ (a,b) ½ ¢ 1/3=1/6 (a,c) ½ ¢ 1/6 = 1/12 (b,a) 1/3 ¢ 1/2=1/6 (b,b) 1/3 ¢ 1/3=1/9 (b,c) 1/3 ¢ 1/6=1/18 E 2 P(E 2 =(i 1 ,i 2 )) (c,a) 1/6 ¢ 1/2=1/12 (c,b) 1/6 ¢ 1/3 = 1/18 (c,c) 1/6 ¢ 1/6=1/36 Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 5 Example (cont’d) Is P(E 2 =(i 1 ,i 2 )) = P(E 1 =i 1 ) ¢ P(E 2 =i 2 ) really a pmf? i1 2 { a,b,c } i2 2 { a,b,c } P(E 2 =(i 1 ,i 2 )) = 1/4+1/6+1/12+1/6+1/9+1/18+1/12+1/18+1/36 = 1 In general, i1,i2,…,in P(E 1 =i 1 ,E 2 =i 2 ,…,E n =i n ) = i1,i2,…,in P(E 1 =i 1 )P(E 2 =i 2 )…P(E n =i n ) = [ i1 P(E 1 =i 1 )][ i2 P(E 2 =i 2 )]…[ in P(E n =i n )] = 1 ¢ 1 ¢ ¢ 1 = 1 Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 6 Example (cont’d) What is the probability that the outcome of the first trial is ‘b’? P(E 1 =b) = P(E 1 =b,E 2 =a) + P(E 1 =b,E 2 =b) + P(E 1 =b,E 2 =c) = 1/6 + 1/9 + 1/18 = 6/18 = 1/3 Monday, April 13, 2009
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03/30/2009 CS206 - Intro. to Discrete Structures II 7 Bernoulli Trial Random experiment with two outcomes (e.g., success & failure), with probabilities p and 1-p . • E.g., flipping a coin rolling 3 dice: success if sum >9, failure if sum 9.
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