I want to get checked if my attempt is okay.

First off, let me shortly describe what Polya's urn is:

A certain urn initially contains a red and a blue ball. We now repeatedly do the following : we (uniformly at random) pick a ball from the urn, and then put it back in together with an additional ball of the same colour. Let $X_n$ denote the number of red balls and $Y_n$ denote the fraction of red balls after we’ve done this $n$ times, i.e. $$ Y_n = \frac{X_n}{n+2}. $$

And the following is my attempt so far. (let me assume that $Y_n$ is a martingale without showing it)

\begin{align*} X_{n+1} &= X_n + 1_{\{\text{$n+1^{th}$ ball is red}\}} \\ X_n &= 1 + \sum_{1\leq i\leq n}1_{\{\text{$i^{th}$ ball is red}\}} \\ \to EX_n &= 1 + \sum_{1\leq i\leq n}E[1_{\{\text{$i^{th}$ ball is red}\}}] = 1 + nE[1_{\{\text{$i^{th}$ ball is red}\}}] = 1 + \frac{n}{2} \end{align*} since the indicator function take only values $0$ or $1$ with the same probability here (picking either ball has the same probability).

Thus, $EX_n = 1 + \frac{n}{2} = \frac{n+2}{2}$, so $EY_n = \frac{1}{2}$.

So now we know the expectation of $Y_n$, but is it enough to say that $Y$ is the uniform distribution on $[0,1]$?

(where $Y_n\xrightarrow[]{a.s.}Y$ (Martingale convergence theorem))

Thank you for your help in advance!

  • $\begingroup$ If you know that $Y_n$ is a martingale, then obviously $\mathbb E[Y_n]=\mathbb E[Y_0]=1/2$. You're more or less proving that $Y_n$ is a martingale again. Also no, it is not enough, $Z_n:=1/2$ is a martingale with expectation $1/2$. $\endgroup$
    – Pierre PC
    Commented Jun 9, 2020 at 23:47
  • $\begingroup$ @PierrePC Thanks for letting me know. But how did you derive that $\mathbb{E}[Y_n] = \mathbb{E}[Y_0] = 1/2$ by the fact that $Y_n$ is a martingale? $\endgroup$ Commented Jun 9, 2020 at 23:51
  • $\begingroup$ @PierrePC Also can you explain what I should show/prove in order to show that $Y$ is uniformly distributed on $[0,1]$? Thank you in advance! $\endgroup$ Commented Jun 9, 2020 at 23:52
  • 1
    $\begingroup$ $\mathbb E[Y_{n+1}] = \mathbb E[\mathbb E[Y_{n+1}|\mathcal F_n]] = \mathbb E[Y_n] = \cdots = \mathbb E[Y_0]$. About the uniform distribution of $Y$ in $[0,1]$, I'd have to think about it but I would use very different methods. $\endgroup$
    – Pierre PC
    Commented Jun 10, 2020 at 0:47
  • 2
    $\begingroup$ In any case, I don't think this question is research level, and I would advise you to post it on math.SE instead. $\endgroup$
    – Pierre PC
    Commented Jun 10, 2020 at 0:48

1 Answer 1


The answer is yes, the distribution of $Y_n$ converges to the uniform distribution on $[0,1]$.
More generally, if we initially have $r$ red and $b$ blue balls in the urn, then the distribution of the proportion of the red balls in the urn converges to the beta distribution with parameters $r,b$; see e.g. Section 4.

Moreover, in your case of $r=b=1$, letting $$p_{n,k}:=P(X_n=k),$$ it is easy to check directly by induction on $n$, using the recursion $$p_{n,k}=p_{n-1,k}\frac{n+1-k}{n+1}+p_{n-1,k-1}\frac{k-1}{n+1}$$ for natural $n$ and $k$, that $$p_{n,k}=\frac1{n+1}\,1\big\{k\in\{1,\dots,n+1\}\big\}$$ for $n=0,1,\dots$; that is, $X_n$ has the discrete uniform distribution on the set $\{1,\dots,n+1\}$.

Therefore, it is clear that the distribution of $Y_n=\frac{X_n}{n+2}$ converges to the uniform distribution on $[0,1]$.


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