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QUESTION A random variable $X$ has pdf $f(x)=(\beta+1)x^{\beta}\ \
0 \leq x \leq 1,\ \ \beta<{-1}$. A random sample of $n$ values of
$X$, $x_1,x_2,...,x_n$ has been obtained.

\begin{description}
\item[(i)]
Find $E(X)$ and use the method of moments to estimate $\beta$.

\item[(ii)]
Show that the likelihood function $L$ is given by

$$\ln L=n \ln(\beta+1)+\beta\sum \ln(x_i)$$

and hence find the maximum estimate of $\beta$.

\end{description}



ANSWER
 $f(x)=(\beta+1)x^\beta\ \ 0 \leq x\leq 1$
  \begin{description}
   \item[(i)]
    $E(X)=\int_0^1(\beta+1)xx^\beta\,dx=[\frac{\beta+1}{\beta+2}x^{\beta
    +2}]_0^1=\frac{\beta+1}{\beta+2}$\\
    Method of moments $\overline{x}=\frac{\hat
    {\beta}+1}{\hat{\beta}+2}\ \
    \overline{x}(\hat{\beta}+2)=\hat{\beta}+1\ \
    \hat{\beta}=\frac{2\overline{x}-1}{1-\overline{x}}$

   \item[(ii)]
   $L=(\beta+1)^n\prod_{i=1}^nx_1^\beta\\
   \ln L=n\ln (\beta+1)+\beta \sum_{i=1}^n \ln (x_i)\\
   \frac{\partial \ln L}{\partial
   \beta}=\frac{n}{\beta+1}+\sum_{i=1}^n \ln (x_i)=0$ when $
   \hat{\beta}=\frac{-n}{\sum_{i=1}^n \ln (x_i)}-1 $( Check this
   is a maximum, $\frac{\partial^2 \ln L}{\partial \beta ^2}<0$
   \end{description}


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