Math 6010 • Linear Models

(P{\'o}lya's Urn) An urn initially contains \(r\) red and \(b\) blue
balls. A ball is chosen uniformly at random (i.e. with probability
\(1/(r+b)\) each). If it comes up red (resp. blue), then it is returned and
another red (resp. blue) ball is added to the urn. The process is repeated
indefinitely. Let \(R_n\) be the number of red balls in the urn after \(n\) draws.
\begin{enumerate}
\item[a)] A vector \((X_1,X_2,\ldots,X_n)\) of random variables is said to be
exchangeable if
\[
  (X_1,\ldots,X_n)\sim(X_{\pi(1)},\ldots,X_{\pi(n)})
  \qquad\text{for all permutations }\pi,
\]
where \(\sim\) denotes equality in distribution. If \(X_i\) is the indicator that
a red ball is drawn from the urn at time \(i\), prove that
\((X_1,\ldots,X_n)\) is exchangeable.
\item[b)] Find the mean and variance of \(S_n=X_1+\cdots+X_n\), the total number
of red balls added to the urn up to time \(n\).
\end{enumerate}