In each case below, the outcome of some function, T ( X ( n ) ) T ( X ( n ) ) , of

In each case below, the outcome of some function, $T\left(X_{\left(n\right)}\right)$, of $n$ iid random variables $X_{\left(n\right)}=(X_1,\dots ,X_n)$ is being considered for providing an estimate of some function of parameters, $q\left(\theta \right)$. Determine whether $E\left(T\left(X_{\left(n\right)}\right)\right)=q\left(\theta \right)$, ${lim}_{n\to \infty }E\left(T\left(X_{\left(n\right)}\right)\right)=q\left(\theta \right)$, and $plim\left(T\left(X_{\left(n\right)}\right)\right)=q\left(\theta \right)$.

(a) $X_i^{\prime }$ 's iid $\mathrm{Gamma}(\alpha ,\beta )$ and $T\left(X_{\left(n\right)}\right)={\overline{X}}_n$ is being used to estimate $q(\alpha ,\beta )=\alpha \beta$.

(b) $X_i$ 's iid $\mathrm{Gamma}(\alpha ,\beta )$ and $T\left(X_{\left(n\right)}\right)={\sum }_{i=1}^n{(X_i-{\overline{X}}_n)}^2/$ $(n-1)$ is being used to estimate $q(\alpha ,\beta )=\alpha {\beta }^2$

(c) $X_i$ 's iid Bernoulli $\left(p\right)$ and $T\left(X_{\left(n\right)}\right)={\overline{X}}_n(1-{\overline{X}}_n)$ is being used to estimate $q\left(p\right)=p(1-p)$.

(d) $X_i^{\prime }$ s iid $N(\mu ,{\sigma }^2)$ and