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Math Help - Distribution of maximal distance between uniformly sampled variables

  1. #1
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    Distribution of maximal distance between uniformly sampled variables

    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!
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  2. #2
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    Re: Distribution of maximal distance between uniformly sampled variables

    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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