For each of the following distributions,find = E( X ), E[ X(X - 1)], a 2 = E[X(X

Question

   For each of the following distributions,find = E( X ), E[ X(X - 1)], a 2 = E[X(X -1)] + E(X) - 2:    a)      f(x)=      3!         (1/4)x (3/4)3-x     x = 0,1,2,3.                      x!(3 - x)!
   b)     f(x)=        4!        (1/2)4     x = 0,1,2,3,4.                       x!(4 - 3)!    For each of the following distributions,find = E( X ), E[ X(X - 1)], a 2 = E[X(X -1)] + E(X) - 2:    a)      f(x)=      3!         (1/4)x (3/4)3-x     x = 0,1,2,3.                      x!(3 - x)!
   b)     f(x)=        4!        (1/2)4     x = 0,1,2,3,4.                       x!(4 - 3)!                       x!(4 - 3)!

Explanation / Answer

(a) X~Binomial (n=3, p=1/4) E(X)=n*p=3*(1/4)=3/4 ^2=n*p*(1-p)=3*(1/4)*(3/4)=9/16 E(X(X-1)) = E(X^2) - E(X) = ^2+ {E(X)}^2 - E(X)=(9/16)+(3/4)^2 - (3/4) = 0.375 (Since ^2=E(X^2)- {E(X)}^2 ) (b) X~Binomial(n=4, p=1/2) E(X)=n*p=4*(1/2)=2 ^2=n*p*(1-p)=4*(1/2)*(1/2)=1 E(X(X-1)) = E(X^2) - E(X) = ^2+ {E(X)}^2 - E(X) =1+(2)^2 - 2= 3 (Since ^2=E(X^2)- {E(X)}^2 )

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