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Suppose R and S are independent and uniformly distributed on (0,1). Let X = R+S and Y = RS. Compute the density of (X,Y), X and Y.
Here is what I have done so far. Since they tell you that X=R+S and Y=RS, this means:
R = Y/S,
S = X-R
and therefore
Y/(X-R) = R
R(X-R) = Y
RX - R^2 = Y
And so
-Y + RX + R^2 = 0
This is equal to R^2 + XR - Y = 0
So now you can use the quadratic formula, with a = 1, b = x, c=-y:
You can have (-x + sqrt(x^2 + 4y))/2
and
(-x - sqrt(x^2 + 4y)/2
The back of my book says that the density is supposed to be
2(x^2 - 4y)^(-1/2), but I'm not sure how to get to that answer from where I left off.
Sorry I meant for that quadratic equation to be R^2 + XR - Y = 0
Therefore a = 1, b = x, c=-y
And so you have (-x + sqrt(x^2 + 4y))/2
or
(-x - sqrt(x^2 + 4y)/2
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