Suppose X1, .. ,XM are independent, identically distributed random variables with mean a a...

Suppose X1, .. ,XM are independent, identically distributed random variables with mean a and variance b2. Let aM ≡ (1/M)Σi=1M aM and bM2≡ (1/(M-1)) Σi=1M (Xi-aM)2. a) Show that aM is an unbiased estimator of E[X]: that is, E[aM] = a. b) Assume that the identity E[ Σi=1M (Xi-aM)2 ] = (M-1) b2 is correct. Show that bM2 is an unbiased estimator of var(X): that is, E[bM2] = b2

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