# Order statistics problem

• Apr 21st 2009, 09:02 AM
dmax
Order statistics problem
I'm not even sure if I can call this an order statistics question, it's kind of an eclectic problem... anyway......

Suppose we have an urn filled with an unknown number W of white balls. We then place a known number y of yellow balls in the urn and mix the contents of the urn vigorously. We then go through the following procedure.

- Remove a ball at random.
- If it is white, then put the ball back in the urn.
- If it is yellow, then we replace the ball with a new white ball. (Note that this means N, the total number of balls in the urn, remains fixed.)
- Repeat until all the yellow balls are removed.

For the following questions, let $T_i$ denote the time until the $i$-th yellow ball is removed. For each of the following we will think of y as a fixed number, while W is a random variable.

a) Determine the distribution of $T_1$ conditioned on the assumption that W = w.

b) Determine the distribution of $T_{i+1} - T_i$ given that W = w.

c) Use the above to calculate $E(T_i | W)$ for $i \in {1, ..., y}.$

d) Take your answer from part (c) and take expectation on both sides of the equation. You should be able to solve the resulting equation for E(W).

e) Discuss intuitively why this procedure allows us to estimate the unknown number of white balls.

My thoughts

Okay, so the number of balls in any urn is y + w. So I think the distribution of $T_1$ in part (a) would be geometric, with parameter ( $\frac{y}{y+w}$). I got the parameter from the chance of picking a yellow ball (y) divided by the number of total balls (y + w). Does that look right?

For (b), would it just be the same as part (a)? Since you could look at $T_{i+1} - T_i$ as being like $T_1 - T_0$, or the number of tries to get a yellow ball after you first start.

For (c), I really don't know how to do this... expected value tends to go right over my head =\

(d) and (e) are dependent on (c) so I'd need some help on (c) before I could get those... any ideas?