PDF of sum of two variables with joint PDF

[vptran84]

Probability Density Function -- Need Help!

Hi,

Can someone please check my work if i did the problem correctly? thanks in advance.

Here is the problem:

Find the PDF of W = X + Y when X and Y have the joint PDF fx,y (x,y) = 2 for 0<=x<=y<=1, and 0 otherwise.

here is my solution:
$$\int_{0}^{1} \int_{0}^{w-y} 2dxdy$$

I work through the integral and get fw (w) = 2w-1 for w>0, and 0 for w<0.

 

[mathman]

Your answer is obviously wrong. f(w) is <0 for w<1/2. Moreover, the integral should be 1 - yours is 0.

 

[vptran84]

ok, i did a little more thinking and this is what i got now...

For region w>0, the region of integration is outside so CDF Fw (w) is 0

For region 0<=w<=1, i used double integration, and i get w^2/2

For region 1<=w<=2, i get 2w-1-w^2/2

For region w>2, i get 1.

So to find PDF, i take the derivative, and i get the following:

fw(w) = w for 0<=w<=1
fw(w) = 2-w for 1<=w<=2
fw(w) = 0 otherwise.

Please let me know if i did anything wrong.

 

[mathman]

Before I looked at your latest post, I worked it out myself. I got the same result as you did.