I am having really hard time with stochastic processes. I will be so glad for your answers. I do not know how to generate the recursive equations.
From an urn containing a white and b black balls, balls are drawn one by one at random according to the following rules:
(i) at any drawing, if the ball drawn is white, then it is returned to the urn,
(ii) if it is black, it is replaced by a white ball (from another collection of balls).
After n such operations, a ball is drawn from the urn. let $W_n$.denote the number of white balls in the urn after the foregoing operations has been repeated n times.
Derive the recursive equation?
ANSWER is E($W_{n+1}$)=(1-1/(a+b))E($W_{n}$)+1
First, compute $P(W_{n+1}=w)$. Notice that total number of balls in the urn always stay unchanged, $a+b$.
We have
$$ P(W_{n+1}=w)=P(W_n=w-1)\left(1-\frac{w-1}{a+b}\right)+P(W_n=w)\frac{w}{a+b} $$
(you could either have $w$ balls and draw a white ball, or $w-1$ white balls and draw a black one so $1$ gets added)
multiply both sides by $w-1$, we have
$$ wP(W_{n+1}=w)-P(W_{n+1}=w)=(w-1)P(W_n=w-1)-\frac{(w-1)^2}{a+b}P(W_n=w-1)+\frac{w^2-w}{a+b}P(W_n=w) $$
Now sum over all possible $w$ (from $0$ to $\infty$)
$$ \mathbb{E}W_{n+1} -1=\mathbb{E}W_n - \frac{1}{a+b}\mathbb{E}W_n^2 + \frac{1}{a+b}\mathbb{E}W_n^2 -\frac{\mathbb{E}W_n}{a+b} $$
So you get
$$\mathbb{E}W_{n+1}=\left(1-\frac{1}{a+b}\right)\mathbb{E}W_n +1 $$
(of course both expectation and second moment exist in this case, since sample space is finite)