(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Two players, A and B, try to see who has the fastest reflects. They test this by using a button, and whoever presses the button first when a light goes on, wins. Let's assume both A and B are random variables ~U(0,1) and are independent (time in seconds). May W = "winner's reaction time" and Z = "loser's reaction time" both random variables.

a) Find the average time for W knowing that the loser took at least 1/2 of a second to press the button.

b) Find the covariance between Z and W.

3. The attempt at a solution

OK, I know that the density for A is f_{A}(a) =1{0<=a<=1} and the one for B is f_{B}(b) =1{0<=b<=1}.

I know that W = min{A,B} and Z=max{A,B}

For point a), what I'm looking for is the average time for W|Z>1/2, which is the same as

(min{A,B}|max{A,B}>1/2). Now, I could find E[W|Z>1/2] if I had its distribution function or its density function. If I tried to find the distribution function, I should look for P(W|_{Z>1/2}<=w) = P(min{A,B}|_{max{A,B}>1/2}<=w). I just don't know what to do with that. And I don't know how I could get the density function alone.

For point b) I think I need the joint probability function for W and Z. And I couldn't find a way to work it out.

Any ideas?

Thanks.

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# Homework Help: Joint probability problems

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