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\begin{document}
\title{Supplementary exercises in STK4011}
%\author{Anders Rygh Swensen \\
%{\footnotesize University of Oslo, Department of Mathematics}
\date{}
%\maketitle
%\footnote{Address: University of Oslo, Department of Mathematics,
%P.O.B. 1053 Blindern, 0316 Oslo, Norway, e-mail: swensen@math.uio.no, ph: +47 28855871 }
%\thispagestyle{empty}%
{\bf \Large Supplementary exercises in STK4011/STK9011}\\

\noindent
{\bf \large Problem 1}\\
Consider the exponential family where the pdf/pmf have the standard form
\[
f(x\vert \mathbf{\eta})=h(x)c^{*} (\mathbf{\eta})  \exp(\Sigma_{i=1}^{k}\eta_{i} t_{i}(x) )
\]
\begin{itemize}
\item[a)] Show that the natural parameter space is convex.
\item[b)] Show that $1/c(\eta)$ is a convex function.
\end{itemize}
\vspace{.5 cm}
{\bf \large Problem 2}\\
Consider the exponential family where the pdf/pmf have the standard form
\[
f(x\vert \mathbf{\eta})=h(x)c^{*} (\mathbf{\eta})  \exp(\Sigma_{i=1}^k\eta_{i} t_{i}(x) )
\]
Prove that,  under the assumption that differentiation and  integration can be interchanged,
\begin{itemize}
\item[a)] $E[t_i(X)] = - \partial/\partial \eta_i \log (c^{*}(\eta)), \;i=1,\cdots,k   $
\item[b)] $Cov(t_i(X),t_j(X)=- \frac{\partial^2}{\partial \eta_i \partial/\eta_j} \log (c^*(\eta)) , \;i,j=1,\cdots,k   $
\end{itemize}
\vspace{.5 cm}
{\bf \large Problem 3}\\
Assume that the random variable X has  pdf
\[
f(x\vert \mathbf{\theta})=(x/\theta^2) \exp(-x^2/2\theta^2 ), \; x>0, \theta>0
\]
which defines the {\em Rayleigh} distribution. Show that the distribution belongs to the exponential family of distributions and compute the expectation and variance of $X^{2}$.

\vspace{.5 cm}
\noindent
{\bf \large Problem 4}\\
Consider the exponential family where the pdf/pmf have the standard  form
\[
f(x\vert \mathbf{\eta})=h(x)c^*(\mathbf{\theta})  \exp(\Sigma_{i=1}^k\theta_{i} t_{i}(x) )
\]
Find an expression for the the moment generating function 
$E[\exp(\Sigma_{i=1}^{k}u_it_i(x))]$,  and verify
that you get the same expectation and variance as in problem 2

\vspace{.5 cm}
\noindent
{\bf \large Problem 5}\\
Consider the  gamma distribution with probability density function
\[ 
\frac{1}{\Gamma(\alpha)\beta^{\alpha}}x^{\alpha -1} \exp(-x/\beta),\; 0<x< \infty,\; \alpha > 0, \beta > 0.
\]
\begin{itemize}
\item[a)]Write the pdf of gamma  distribution on the standard form and find the natural
parameters $\eta_1$ and $\eta_2$ and the statistics $T_1$ and $T_2$. 
\item[b)]
Express the
expectations of $\log (X)$ by the natural parameter and vice versa. 
\end{itemize}

\vspace{.5 cm}
\noindent
{\bf \large Problem 6}\\
Let X and Y be independent, $X \sim U[0,1] $ and $Y \sim U[0,1] $. 
Find the pdf, probability density function,  and mgf,  moment generating
function, of $X+Y$.  


\vspace{.5 cm}
\noindent
{\bf \large Problem 7}\\
Consider a discrete sample  $\mathbf{X}$ with probability mass function $f(\mathbf{x}\vert \theta)$.
Show that if $T(\mathbf{X})$ is a sufficient and complete statistic and $U(\mathbf{X})$ is a 
minimal sufficient statistic, then $P(U(\mathbf{X})=U(\mathbf{x}))=P(T(\mathbf{X})=T(\mathbf{x}))$
for all $\mathbf{x}$ such that $P(\mathbf{X}=\mathbf{x}) >0$.



















\end{document}
