CT3 IAI May 2014 Q4

Let X and Y be two independent random variables. Define V = Max (X, Y) and W = Min (X, Y). Let FX, FY, FV and FW denote the cumulative distribution functions of X, Y, V and W respectively



Now I understand Fv(t)= P(V<=t)
=P(MAX(X,Y)<=t)
in the next step why does the function max disappears and we get P(X<=t and Y<=t)
We should get P(MAX(X<=t,Y<=t)

 

We have P[(Max. X,Y)≤t]
Now, think if the max. of X and Y is less than t then obviously the other value will also less than t.
{Eg. Suppose X=4, Y=5 and t=7
Max(4,5)<7 then both values are less than 7}

So..P[(Max.X,Y)≤t] = P(X≤t)×P(Y≤t)
(since X,Y are independent)
= Fx(t)*FY(t)

 

Thanks