An urn initially contains r red and g green balls. A ball is chosen at random from the balls find the marginal probability 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.
 A: $P\left[X_{1}=1\wedge X_{2}=1\right]=P\left[X_{2}=1\mid X_{1}=1\right]P\left[X_{1}=1\right]=\frac{r+c}{r+g+c}\frac{r}{r+g}$
$P\left[X_{1}=1\wedge X_{2}=0\right]=P\left[X_{2}=0\mid X_{1}=1\right]P\left[X_{1}=1\right]=\frac{g}{r+g+c}\frac{r}{r+g}$
$P\left[X_{1}=0\wedge X_{2}=1\right]=P\left[X_{2}=1\mid X_{1}=0\right]P\left[X_{1}=0\right]=\frac{r}{r+g+c}\frac{g}{r+g}$
$P\left[X_{1}=0\wedge X_{2}=0\right]=P\left[X_{2}=0\mid X_{1}=0\right]P\left[X_{1}=0\right]=\frac{g+c}{r+g+c}\frac{g}{r+g}$
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.
