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5. Let X,Y ind G(p). Use the expectation method, in combination with the indicator function method, to find an unbiased estim

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Define~Z=1~if~X=0,Y=0\\ =0~otherwise\\\\ Then~E(Z)=1*P(X=0,Y=0)+0*(1-P(X=0,Y=0))\\\\ =P(X=0,Y=0)=P(X=0)P(Y=0)~(Since~X~and~Y~are~independent)\\\\ =p^2\\\\ where,~pmf~of~X~is\\ P(X=x)=p(1-p)^{x-1};~x=1,2,....\\ =0~otherwise\\\\ pmf~of~Y~is\\ P(Y=y)=p(1-p)^{y-1};~y=1,2,....\\ =0~otherwise\\\\ Hence~Z~is~an~unbiased~estimator~of~p^2.

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