4. In each case below, the outcome of some function, T(X(n)), of n iid random variables X(n)(X1,...,

4. In each case below, the outcome of some function, T(X(n)), of n iid random variables X(n)¼(X1,..., Xn) is being considered for providing an estimate of some function of parameters, q(y). Determine whether E(T(X(n))) ¼ q(y), limn!1E T XðnÞ

¼ qð Þy , and p lim TðXðnÞÞ ¼ qðyÞ.

(a) Xi’s ~ iid Gamma(a,b) and T(X(n)) ¼ X n is being used to estimate q(a,b)¼ab.

(b) Xi’s iid Gamma(a,b) and TðXðnÞÞ ¼ Pn i¼1 Xi  X n 2

/

(n1) is being used to estimate q(a,b)¼ab2

(c) Xi’s ~ iid Bernoulli (p) and TðXðnÞÞ ¼ X nð1  X nÞ is being used to estimate q(p) ¼ p(1p).

(d) Xi’s iid N(m,s2

) and TðXðnÞÞ¼ðPn i¼1 Xi  n1=2Þ=ðn þ 1Þ

is used to estimate m.