Combinatorics and/or Statistics
Maths and statistics discussion, revision, exam and homework help.
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Combinatorics and/or StatisticsThere are N fish in a lake, of which an unknown number Q are diseased. The prior distribution of Q is discrete uniform on {0,1,...,N}. Let R denote the number of diseased fish in a random sample of n fish
Show that![\[
{\mathop{\rm P}\nolimits} (Q = q|R = r) = \frac{{\left( {\begin{array}{*{20}c}
q \\
r \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{N - q} \\
{n - r} \\
\end{array}} \right)}}{{\left( {\begin{array}{*{20}c}
{N + 1} \\
{n + 1} \\
\end{array}} \right)}}
\]](http://www.thestudentroom.co.uk/latexrender/pictures/c1/c1014fb7444e30aa63dd2662221d6362.png "\[
{\mathop{\rm P}\nolimits} (Q = q|R = r) = \frac{{\left( {\begin{array}{*{20}c}
q \\
r \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{N - q} \\
{n - r} \\
\end{array}} \right)}}{{\left( {\begin{array}{*{20}c}
{N + 1} \\
{n + 1} \\
\end{array}} \right)}}
\]")
where![\[
r \le q \le N - n + r
\]](http://www.thestudentroom.co.uk/latexrender/pictures/45/45638a7f032d0b22333bd24b7fef6605.png "\[
r \le q \le N - n + r
\]")
^^(I've done this bit)
Find the posterior expectation of Q given R=r.
I've tried to do this last bit via some kind of complicated combinatorics but can't get anywhere - I feel I'm missing some simple trick but can't think what. Any help would be much appreciated!Last edited by thatguyoverthere; 18-01-2012 at 16:42. Reason: Did the question.
