Distribution of Number of Pieces

[mXSCNT]

Let $$\{A_i\}$$ be independent random variables, real numbers selected uniformly from the interval (0,1-v) for some constant v, 0<v<1.
Let $$B_i = \cup^i_{j=1} (A_j,A_j+v)$$
Let $$C_i$$ be the number of disconnected pieces of $$B_i$$.

Problem: What is the distribution of $$C_i$$? I doubt that a closed form expression is possible but it's tough to even find a computer program to calculate it except via monte carlo.

 

[techmologist]

Let $$\{A_i\}$$ be independent random variables, real numbers selected uniformly from the interval (0,1-v) for some constant v, 0<v<1.
Let $$B_i = \cup^i_{j=1} (A_j,A_j+v)$$
Let $$C_i$$ be the number of disconnected pieces of $$B_i$$.

Problem: What is the distribution of $$C_i$$? I doubt that a closed form expression is possible but it's tough to even find a computer program to calculate it except via monte carlo.

I haven't worked out the details, but I think I see a way to approach this. Even if it doesn't result in a closed form formula, you might be able to work out an algorithm to compute it. Basically, the hard part is finding the volume left of an i-cube of side 1-v after several pieces have been sliced off. It should be easier for v >= 1/3, but tricky for v < 1/3. The idea is to work with the order statistics of the A_j, and find the probability that there are exactly k-1 "gaps" of the form A_(m+1)-A_(m) > v. In this case C_i = k. Is that making sense?

Mind if I ask what is the application of this?

 

[techmologist]

Conjecture:

$$P\{C_i = k\} = \frac{1}{(1-v)^i}\binom{i-1}{k-1}\sum_{j=0}^{k-1}(-1)^j\binom{k-1}{j}G(i-k+j, i; v)$$

where

$$G(m,i;v) = \sum_{n=1}^{r}(-1)^{n+1}\binom{m}{n-1}(1-nv)^i$$

for

$$\frac{1}{r+1} \leq v < \frac{1}{r}$$