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I'm self studying statistics, and this is one hw problem that I'm getting stuck at.

Suppose in two independent trials with prob p of success, we observe X successes. Find an unbiased estimator T(X) of the function $g(p)=p^2$

Is it simply $(X/2)^2$?

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up vote 2 down vote accepted

Nonlinear transformations generally do not preserve unbiasedness of estimators. Explicitly calculate: $E[(X/2)^2]=(1-p)^2 \times (0/2)^2 + 2p(1-p) \times (1/2)^2 + p^2 \times (2/2)^2 = (p+p^2)/2 \neq p^2$. One unbiased estimator would be $(X^2-X)/2$.

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