Let X and Y be independent continuous random variables that are U(-1,1) (i.e., they have uniform distribution over the interval (-1,1)). Note that this implies that their CDFs are Fx(x)=(1/2)(x+1) for -1 < x < 1 and FY(y)=(1/2)(y+1) for -1 < y < 1 (These are also equal to 0 over an appropriate interval and 1 over another appropriate interval).
Let V = max(X,Y) and W= 1-2x2 (a) Show work by hand to confirm that the PDF of V is fV(v)=(1/2)(v+1) for -1 < v < 1 (and it is zero otherwise). Start your work by finding the CDF FV(v). (b) Show work by hand to confirm that the mean is v = E[V] = (1/3). (c) Show work by hand to confirm the PDF of W is for -1 < w < 1 (and it is zero otherwise). Start your work by finding the CDF of FW(w). (d) Show your work by hand to confirm that the mean is W = E[W} = (1/3). You may need to do a substitution to help you do an appropriate integral here.
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