# Statistics Problem: finding a PDF using the CDF technique

Hey guys, I'm stuck on a question in my homework assignment and I was wondering if you could push me in the right direction? So here's the question:

X and Y are continuous random variables with joint pdf f(x,y)= 4xy (0<x<1, 0<y<1, and otherwise 0). Find the pdf of T=X+Y using the CDF technique.

So this is how I started off, I first let G(t) be the CDF of T, then I look at three different cases:
t<=0: G(t) = P[X+Y<=t] = 0
t>=2: G(t) = P[X+Y<=t] = 1
0<t<2: G(t) = P[X+Y<=t] = ?
So here I'm a little confused, I'm trying to figure out the limits of the double integral I'm supposed to take, I think I might have to take one integral from 0<t<1, then another from 1<=t<2, but then I'm stuck with two "functions" (one from (0,1), one from [1,2)) for my g(t). Are there any possible limits for this double integral that would save me from having to separate 0<t<2 into two double integrals?

Ray Vickson
Homework Helper
Dearly Missed
Hey guys, I'm stuck on a question in my homework assignment and I was wondering if you could push me in the right direction? So here's the question:

X and Y are continuous random variables with joint pdf f(x,y)= 4xy (0<x<1, 0<y<1, and otherwise 0). Find the pdf of T=X+Y using the CDF technique.

So this is how I started off, I first let G(t) be the CDF of T, then I look at three different cases:
t<=0: G(t) = P[X+Y<=t] = 0
t>=2: G(t) = P[X+Y<=t] = 1
0<t<2: G(t) = P[X+Y<=t] = ?
So here I'm a little confused, I'm trying to figure out the limits of the double integral I'm supposed to take, I think I might have to take one integral from 0<t<1, then another from 1<=t<2, but then I'm stuck with two "functions" (one from (0,1), one from [1,2)) for my g(t). Are there any possible limits for this double integral that would save me from having to separate 0<t<2 into two double integrals?

You are correct in that you do need different integrals for t > 1 and t < 1. If you use the cdf method this is unavoidable. (Other, completely different, methods are easier, but you are told not to use them.)

RGV

You are correct in that you do need different integrals for t > 1 and t < 1. If you use the cdf method this is unavoidable. (Other, completely different, methods are easier, but you are told not to use them.)

RGV

Oh, okay, so how do I get a g(t) with only one function from 0<t<2 instead of having:
g(t)={f(t), t<1
h(t), t>1)
?

Does it make sense to write:
G(t)={0, t<0
F(t), t<1
H(t), t>1
1, t>2}
as:
G(t)={0, t<0
F(t)+H(t), 0<t<2
1, t>2}
?

EDIT: Nevermind, I think I was doing the other part of the question wrong.. thanks for your help!

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