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Polya’s urn model supposes that an urn initially contains $r$ red and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with m other balls of the same color. Let $X_k$ be the number of red balls drawn in the first $k$ selections.

(a) Find $\mathbb{E}[X_1]$.

(b) Find $\mathbb{E}[X_2]$.

(c) Find $\mathbb{E}[X_3]$.

(d) Conjecture the value of $\mathbb{E}[X_k]$, and then verify your conjecture by a conditioning argument.

(e) Give an intuitive proof for your conjecture.

tortue
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Leona Wu
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1 Answers1

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Clearly, for the distribution of $X_1$ is Bernoulli with parameter $p$ being equal to $\frac{r}{b + r}$ and hence $$ \mathbb{E} X_1 = \frac{r}{b + r}. $$ In fact, the answer remains the same for $\mathbb{E} X_n, n > 1$. See this link for detailed intuitive explanation.

P.S. I'm also marking this question as a duplicate due to the link provided above.

tortue
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