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**Two "Sum of Random Variables" Problems**

## Homework Statement

*Problem A:*

Consider two independent uniform random variables on [0,1]. Compute the probability density function for Y = X

_{1}+ 2X

_{2}.

*Problem B:*

Edit: never mind, solved this one

## Homework Equations

f

_{Y}(y) = F'

_{Y}(y)

F

_{Y}(y) = double integral (X

_{1}+2X

_{2}<= y) f

_{X1X2}(x

_{1},x

_{2})

f

_{X1X2}(x

_{1},x

_{2}) = f

_{X1}(x

_{1})*f

_{X2}(x

_{2}) since they're independent

for uniform RVs on [a,b], f

_{X}(x) = 1/(b-a) = 1 in this case

## The Attempt at a Solution

We worked the Y=X

_{1}+X

_{2}version of this in class, so I feel like I at least have some idea where to start. Since X only exists inside [0,1], I know I have to break that implicit integral down into several ranges. It looks like I need to break it down into 0<y<1, 1<y<2, and 2<y<3. I can reduce the 0<y<1 case to integral(0 to y/2) dx

_{2}* integral(0 to y-2x

_{2}) dx

_{1}.

My problem is the other two ranges. I know I can express the 2<y<3 range as 1 - the triangle above the y=X

_{1}+2X

_{2}line, I'm just not sure how to describe that triangle. As for the 1<y<2 range, that's not even a triangle so I'm not sure how to explicitly set the limits of integration; don't I need to break it up into 2 different integrals? We haven't worked any examples like that in class, and it's been over ten years since I took calculus, so...

Last edited: