Independent observations X1......Xn are such that Xi belongs to {0.,1} and for some 0<theta<1, it is that Pr(Xi=1)=theta.
Let n1 be the number equal to 1.
Find the expected value and variance of n1?
Let's use k instead of and p instead of because they are 'more familiar'. The probability to have k ones among n variables is...
(1)
... where...
(2)
The expected value of k is then...
(3)
... and the variance...
(4)
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