RecitationEightsols - Rensselaer Electrical, Computer, and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 8 Solutions Wednesday April 7, 2010 CommonMeansandVar iances —d iscreteRVs Find the means and variances of the following discrete random variables. family pmf mean μ mean square variance σ 2 Bernoulli P B ( k )= ½ 1 ,p 0 ,q p p pq Binomial b ( k ; n, p ¡ n k ¢ p k q n k np ( np ) 2 + npq npq geometric 1 1+ μ ³ μ 1+ μ ´ k u ( k ) μ μ +2 μ 2 μ + μ 2 Poisson ( λT ) k k ! e λT u ( k ) λT ( λT ) 2 + λT λT q , 1 p Note: The geometric pmf is sometimes written in terms of the parameter a as (1 a ) a k u ( k ) with 0 <a< 1 . Then μ = a/ (1 a ) with μ> 0 . Some useful de f nitions: mean-square value or average power : E [ X 2 ] : σ 2 = E [( X μ ) 2 ] = E [ X 2 2 + μ 2 ] = E [ X 2 ] 2 μE [ X ]+ μ 2 = E [ X 2 ] μ 2 , since E [ X ]= μ. 1. ( Bernoulli ) Ans : μ = 1 X k =0 kP B ( k ) =0 q +1 p = p. E [ X 2 1 X k =0 k 2 P B ( k ) 2 q 2 p = p, 1
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and so σ 2 = E [ X 2 ] μ 2 = p p 2 = p (1 p )= pq. 2. ( Binomial ) Ans: μ = n X k =0 kb ( k ; n, p ) = n X k =0 k μ n k p k q n k = n X k =1 k n !
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RecitationEightsols - Rensselaer Electrical, Computer, and...

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