\frametitle {Examples of likelihood functions}Poisson counts\\
\begin{itemize}
\item We observe the number of phone calls at various calling centres over a given period and denote them by
$(x_1,...,x_n)$.
\item  We assume that the $x_i$ are \underline{independent} realizations of a Poisson random variable $X_i$ with parameter $\lambda$, i.e.
\pause $P(X_i=x) \pause = \pause \exp(-\lambda) \lambda^{x}/x!$ \\
\pause \item  NB: $x \in \mathbb{N}$ and $\lambda \in \mathbb{R}_+$
\pause \item  $E[X_i] = \lambda$ and $V[X_i] = \lambda$
\end{itemize}
%% \pause
%% {\scriptsize
%% \begin{block}{Examples of Poisson probability mass functions:}
%% \begin{verbatim}
%% lambda = 5
%% xx=0:(3*lambda)
%% plot(xx,dpois(xx,lambda=lambda),type='n',
%%      xlab='x',ylab='Probability mass',sub=bquote(lambda==.(lambda)),cex.sub=1.4,
%%      main='Poisson probability mass function')
%% points(xx,dpois(xx,lambda=lambda),
%%        type='h',lwd=10,col='orange')
%% \end{verbatim}
%% \end{block}
%% }