An Introduction to Probability and Statistical Inference, 2nd Edition

Section 9.5

5.2 

(i) $\tilde{\theta }=\left(1-2\overline{X}\right)/\left(\overline{X}-1\right);$ (ii)$\tilde{\theta }=3$ and $\widehat{\theta }\simeq \mathrm{3.116.}$

5.4 Follows from Exercise 3.15(i).

5.6 

(i) The EX and EX² follow by integration, and Var(X) = ß².

(ii) $\tilde{\alpha }=\overline{X}-S,\tilde{\beta }=S,$ where $S^2=\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{\left(X_i-\overline{X}\right)}^2.}$

5.8 By equating the second-order sample moment ...