probability question,interval [0,1] is split to 3 parts by 2 random points X,Y~U[0,1]. X,Y are independent. Z=min' of all 3 parts. find E(Z) I am having trouble solving this question. I saw a solution that used a bit of geometry (plotted Y vs. X axis and used areas to found F(z), the CDF of the Min' value) but I want to know if there is another way of approaching questions like this.
$X,Y$ are independent and uniformly distributed $U[0,1]$
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The interval is splitted to parts $ [X,Y-X,1-Y]$ ...
thanks!
 A: Problems like this can be easily solved using a computer algebra system. For your example, by independence, random variables $X$ and $Y$ have joint pdf $f(x,y)$:
f = 1;   domain[f] = {{x, 0, 1}, {y, 0, 1}};

Then, the solution to your problem can be obtained as a simple one-liner as:
Expect[Min[x, y-x, 1-y], f]


-1/9

where Expect is a function from the mathStatica add-on to Mathematica. Similar packages exist for Maple. 
If your problem is to show 'all the steps manually', solve the integrals etc as a homework exercise ..., then I am not sure that it is appropriate for others to fulfil that role here. But at least you have the answer to work towards.   

In response to the suggestion below by robjohn, the Expect function (which is part of the mathStatica package of which I am one of the authors) calculates symbolic / theoretical solutions to arbitrary expectation problems, given an arbitrary user-specified pdf $f(x)$ - whether continuous, discrete, multivariate or univariate etc, and it does so by taking advantage of a computer algebra system (in this case Mathematica). The solutions produced are exact symbolic solutions - not numerical nor approximate. I believe similar functionality exists for Maple (I am not an author of that software). We have received requests to build a version of mathStatica for Maple, but there is no plan to do so at the present time.
