please help with stats question

Question

Suppose that X~ B(12,p) and consider the point estimate,

 

p-hat = X/14

 

a) What is the bias of this point estimate?

b) What is the variance of this point estimate?

c) Show that this point estimate has a mean square error of (3p - 2p^2)/49

Answer 1

Note that Bias(estimator,parameter) = E(estimator) - parameter. The mean squared error is the second-moment centered around the population parameter.

Bias(p-hat,p) = Bias(X/14,p) = E(X/14) - p = 12p/14 - p = -p/7

Var(p-hat) = Var(X/14) = Var(X)/196 = 12p(1-p)/196 = 3p(1-p)/49

MSE(p-hat,p) = E[(X/14 - p)^2] = E(X²)/196 - pE(X)/7 + p² = [Var(X) + E(X)²]/- pE(X)/7 + p²

= [12p(1-p) + 144p²]/196 - 12p²/7 + p² = (12p - 12p² + 144p²)/196 - 12p²/7 + p²

= 3p/49 + 33p²/49 - 12p²/7 + p² = 3p/49 - 2p²/49 = (3p - 2p²)/49