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

f X,Y(x,y) = (8 +xy^3)/64, if -1<x<1, -2<y<2

0, otherwise

Find the probability density function of W = 2X+Y.

2. Relevant equations

F(w) = Pr{W≤w}=∫∫f(x,y)dxdy

f(w) = d/dw F(w)

3. The attempt at a solution

I found the support of W to be -4<w<4

I think the region should be broken up into: F(w) = 0, w<-4

A, -4≤w<0

B, 0≤w<4

1, 4≤w<∞

For expression B, I took the double integral: ∫-2 to 2 ∫ -1 to (w-y)/2 [(8+xy^3)/64]dxdy and got (w+5)/10.

The x-bound (w-y)/2 comes from the given 2X+Y=W, isolating X

For expression A, I tried the double integral ∫-2 to (w+2)∫-1 to (w-y)/2 [(8+xy^3)/64]dxdy, but am finding myself unsure of this setup (the result is a very lengthy integral). The y-bound of w+2 comes from the top intersection point of the line 2X+Y=W and x=-1.

Is there some flaw in my logic here? Any input on any part of this is much appreciated...I'm not positive about any of it, so if you see a flaw somewhere, please let me know. Also, just to clarify, what I am doing is trying to find the cdf which I can then differentiate to get the pdf. There may be other ways, but this is the only method we have covered thus far in class, so if we can stick to that, it would be great! Thanks!

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# Homework Help: Continuous Variable pdf from a cdf

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