# Using MGF for determining distribution of a statistic

Suppose $X_1;X_2;\cdots;X_n$ are iid Weibull(1;$\beta$).

1) Find the mgf of $\bar{X}$.

2) Use the mgf to and E[$\bar{X}$].

3) Find the VAR[$\bar{X}$] by any method.

4) Suppose the parameter $\beta$ now has a gamma($\alpha$,$\theta$) distribution. Find E[$\bar{X}$] and VAR[$\bar{X}$].

5) Finally, suppose the parameter $\theta$ in the distribution of $\beta$ has a lognormal($\mu, \sigma^2$) distribution. Find E[$\bar{X}$].

For 1-3, I took the moment generating functions and multiplied them to get $[E(X^n)]^n = [B^n\Gamma(1+n)]^n$. My n is given as 1 for each Weibull so I have $[\beta^1]^n=\beta^n$. For the Variance, I took $E(\bar{X^2}) - [E(\bar{X})]^2 = 2\beta^{2n} - \beta^{2n} = \beta^{2n}$.

For 4, I take $E(\bar{X}) = E[E(\bar{X}|\beta)] = E(\beta^n) = \beta^n$. Then $V(\bar{X}) = E[V(\bar{X|\beta})] + V[E(\bar{X}|\beta) = E(\beta^{2n})+ V(\beta^n)$. This doesn't seem right thought. I am also getting similar stuff for 5. Any help is greatly appreciated.

For 5,

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