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An urn initially contains $r$ red and $g$ green balls. A ball is chosen at random from the balls in the urn and its colour is noted. Then it, together with $c > 0$ balls of the same colour as the drawn ball, are added to the urn. The procedure is repeated $n - 1$ times so the total number of drawings made from the urn is $n$. For $i=1,2,....,n$ let $X_i$ be the random variable that takes the value $1$ if the $i$-th ball drawn is red and takes the value $0$ if this ball is green. Find the marginal distributions of $X_1$ and $X_2$, and hence find the mean and variance of these two random variables. Find $Cov(X_1,X_2)$. Assume $P(X_i = 1) = r/(r + g)$ for all $i$. Hence find $E(S)$, where $S$ is the total number of red balls drawn in the $n$ drawings.

Could anyone please help me with this question, as I have got more questions similar to this. If I know the method of doing this, I would be able to deal with the other questions on my own.

Kasun Jayasuriya
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$P\left[X_{1}=1\wedge X_{2}=1\right]=P\left[X_{2}=1\mid X_{1}=1\right]P\left[X_{1}=1\right]=\frac{r+c}{r+g+c}\frac{r}{r+g}$

$P\left[X_{1}=1\wedge X_{2}=0\right]=P\left[X_{2}=0\mid X_{1}=1\right]P\left[X_{1}=1\right]=\frac{g}{r+g+c}\frac{r}{r+g}$

$P\left[X_{1}=0\wedge X_{2}=1\right]=P\left[X_{2}=1\mid X_{1}=0\right]P\left[X_{1}=0\right]=\frac{r}{r+g+c}\frac{g}{r+g}$

$P\left[X_{1}=0\wedge X_{2}=0\right]=P\left[X_{2}=0\mid X_{1}=0\right]P\left[X_{1}=0\right]=\frac{g+c}{r+g+c}\frac{g}{r+g}$

This offers you the distribution of $\left(X_{1},X_{2}\right)$ and things like $\text{cov}\left(X_{1},X_{2}\right)$ can be calculated on base of it.

Note that $P\left[X_{2}=1\right]=P\left[X_{1}=1\wedge X_{2}=1\right]+P\left[X_{1}=0\wedge X_{2}=1\right]=\frac{\left(r+c\right)r+rg}{\left(r+g+c\right)\left(r+g\right)}=\frac{r}{r+g}=P\left[X_{1}=1\right]$ This tells you about the marginal distributions of $X_{1}$ and $X_{2}$. Both have the Bernoulli distribution with parameter $p=\frac{r}{r+g}$.

Note that $S=X_{1}+...+X_{n}$ making it possible to determine its expectation on base of the expectations of the $X_{i}$ and here you are asked to assume (not to prove) that all the $X_{i}$ have the same distribution.

drhab
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  • Please let me know the in detail procedure and the answers to this questions. – Kasun Jayasuriya Nov 18 '13 at 16:27
  • I added some things to my answer. If you don't understand something , then tell me exactly what you don't understand. – drhab Nov 18 '13 at 17:59
  • Are you sure that its Bernoulli distribution? since it has n trials I thought its Binomial distribution. – Kasun Jayasuriya Nov 18 '13 at 18:57
  • The $X_{i}$ can only take values in set $\left{ 0,1\right} $ (red of not red). This is characteristic for a Bernoulli distribution. It can be ( I haven't checked that) that $S$ is binomially distributed. But that is not relevant here. You are only asked to determine its expectation wich is possible allready on base of $S=X_{1}+...+X_{n}$. We have $E\left[S\right]=E\left[X_{1}\right]+\cdots+E\left[X_{n}\right]$ – drhab Nov 18 '13 at 19:13