In the urn we have two balls: red and green. In each step we draw uniformly one of the balls, we check its colour, return it to the urn and than we are adding to the urn the next ball of the same colour. Let $M_n$ be the ratio of red balls to all balls in the step $n$. Of course $M_0=\frac12$.
Problems
- Show that $\{M_n\}_{n\geq0}$ is a matringale.
- Show that for every $n$, distribution of $M_n$ is uniform on the set $\{\frac{1}{n+2}, \frac{2}{n+2},\ldots,\frac{n+1}{n+2}\}$ (proof by induction).
- Justify that the sequence $\{M_n\}_{n\geq0}$ is convergent in distribution and determine its limit.
My attempts
- We define the random variable $Y_n$. Let $Y_0=1$ and for $n>1$: $Y_n=1$ if in $n$-th drawing we have drawn the red ball and $Y_n=0$ if green. Let $r_{n+1}=r_n+Y_{n+1}$, $P(Y_{n+1}=1|Y_1,\ldots,Y_n)=M_n$.
$\operatorname{E}[M_{n+1}|M_n]= \operatorname{E}\left[\dfrac{r_{n+1}}{r_{n+1}+g_{n+1}}|M_n\right]= \operatorname{E}\left[\dfrac{r_n+Y_{n+1}}{r_{n}+g_{n}+1}|M_n\right]= \operatorname{E}\left[\dfrac{r_n}{r_{n}+g_{n}+1}|M_n\right]+\operatorname{E}\left[\dfrac{Y_{n+1}}{r_{n}+g_{n}+1}|M_n\right]= \dfrac{r_n}{r_{n}+g_{n}+1}+\dfrac{1}{r_{n}+g_{n}+1}\operatorname{E}\left[Y_{n+1}\right]= \dfrac{r_n}{r_{n}+g_{n}+1}+\dfrac{1}{r_{n}+g_{n}+1}M_n= \dfrac{r_n}{r_{n}+g_{n}+1}+\dfrac{r_n}{(r_{n}+g_{n}+1)(r_n+g_n)}= \dfrac{r_n(r_n+g_n)}{(r_{n}+g_{n}+1)(r_n+g_n)}+\dfrac{r_n}{(r_{n}+g_{n}+1)(r_n+g_n)}= \dfrac{r_n(r_n+g_n+1)}{(r_{n}+g_{n}+1)(r_n+g_n)}= \dfrac{r_n}{r_n+g_n}=M_n. $
I would really appreciate any help with the second and the third problem, because I have no idea how to begin.
