## Polya’s Urn

At time 0, an urn contains 1 black and 1 white ball. At each time 1,2,3,…, a ball is chosen at random from the urn and is replaced together with a new ball of the same colour. Just after time n, there are therefore n+2 balls in the urn, of which B_{n}+1 are black, where B_{n} is the number of black balls chosen by time n.

Let M_{n}=(B_{n}+1)/(n+2), the proportion of black balls in the urn after time n. Prove that (relative to a natural filtration which you should specify) M is a martingale.

Prove that P(B_{n}=k) = (n+1)^{-1} for 0 \leq k \leq n . What is the distribution of \mathbb{\Theta} , \textit{where } \mathbb{\Theta}:= \lim M_n ?

Prove that for 0 < \theta < 1 ,

N_n^\theta : = \frac{(n+1)!}{B_n! (n-B_n)!} \theta^{B_n} (1-\theta)^{n-B_n}

defines a martingale.