Question: Let T = n k=1 kX k , where the X k are independent randomvariables wit

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Question: Let T = nk=1kXk, where the Xk are independent randomvariables with means and variances 2. Find E(T) and Var(T). What I know: the constant infront of Xk isbk= k.                                                  n I also need to use the results of k2 =n(n + 1)(2n + 1) / 6                                                 k=1 Question: Let T = nk=1kXk, where the Xk are independent randomvariables with means and variances 2. Find E(T) and Var(T). What I know: the constant infront of Xk isbk= k.                                                  n I also need to use the results of k2 =n(n + 1)(2n + 1) / 6                                                 k=1

Explanation / Answer

Given T = nk=1 kXknow E(T) = E(nk=1 kXk)since k are constants we can take the expectation inside overXk thus        =nk=1 k E(Xk) =nk=1 k taking common        = nk=1 k as sum on n natural number isn*(n+1 )/2       = *n *(n+1)/2 now E(T2) = E(nk=1k2X2k + cross term havingproduct between X and Xb such that not equal to )                   =E(nk=1k2X2k ) + 0 becauseX and Xb are independent is not equal to hence E() will be 0                   =nk=1k2 E(X2k )                   =nk=1k2 2                   =2 k2 which is given                   =  2n(n + 1)(2n + 1) / 6                   =  2n(n + 1)(2n + 1) / 6

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