Distribution of maximal distance between uniformly sampled variables

**quoth** · July 10th 2011, 04:18 PM

Hi everyone!

I've got $n$ uniformly distributed iid random variables $X_i\backsim U(0,1)$ .

Then I sort them ( $i>j \rightarrow X_i > X_j$ ).
My interest lies in

$Y = X_{i+1}-X_{i}$

the distance between two adjacent values.

What I'd like to know is the distribution of $Y$ . I'm working on a delta-encoding method (storing a sorted list by their differences indead of directly storing the values) and this would help me understand what values I can expect. So a decent approximation is good enough for me.

Thanks a lot!

**Moo** · July 12th 2011, 02:08 PM

Hello,

Well I don't know about the approximation, but it's easy to get the pdf of Y if you're familiar with the change of variable transform for the pdf.
Otherwise, you could also do it this way, with conditional expectations (your question doesn't look like high school probability) :

it would approximately give this, with a suitable y... :

$P(Y<y)=P(X_{i+1}-X_i<y)=E[P(X_{i+1}-X_i<y | X_i)]=E[P(X_{i+1}<y+X_i | X_i]=E[y+X_i]=y+1/2$

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