Originally Posted by

**Laurent** (in fact, this theorem is the exact analog to the theorem about ordered uniform random variables in the 1-dimensional case)

You could find the distribution of $\displaystyle Y_2$ given $\displaystyle N(1)=10$. To that aim, for any $\displaystyle 0<r<1$, you have:

$\displaystyle P(Y_2> r, N(1)=10)= P(N(r)\leq 1, N(1)=10)=$ $\displaystyle P(N(r)=0, N(1)=10) + P(N(r)=1, N(1)=10)$

and all these probabilities can be computed easily using annuli. For instance, $\displaystyle P(N(r)=1,N(1)=10)$ is the probability that there is 1 point in $\displaystyle D(0,r)$ and 9 points in $\displaystyle D(0,1)\setminus D(0,r)$, and since these sets are disjoint, the numbers of points therein are independent (and Poisson distributed with parameter proportional to the area...).

Finally, you can use $\displaystyle E[Y_2|N(1)=10]=\int_0^1 P(Y_2>t|N(1)=10)dt$ to compute the expectation.

I let you try that.