Let $X$ and $Y$ be two random variables i.i.d $U(0,1)$. Find the joint pdf of $T = X+Y$ and $U = \frac{X}{X+Y}$ and the marginal densities of $T$ and $U$

My attempt:

We will have the following transformation:

$X = TU$ and $Y = T - TU$. The jacobian is $J = -ut -t(1-u) = -t$ and the joint pdf:

$$f_{T,U}(t,u) = I_{(0,1)}(tu)I_{(0,1)}(t-tu) |t|$$

where $I$ is the indicatr function

Note that $U$ and $T$ will be jointly defined in the following region:

1) $0 < t < 2$

2) $0 < u < \infty$

3) $0 < tu < 1 \Rightarrow t<\frac{1}{u} (u>0)$

4)$ 0 < t -tu < 1$

4.1) $u < 1 (t>0)$

4.2) $t < \frac{1}{1-u}$

Integrating in respect of $t$, I could obtain th right pdf of $U$ because I know the answer. But in respect of $u$ I couldn't. Is my region wrong?

P.S: I know how to obtain the pdf of $X+Y$ using other ways, I want it using this joint distribution.



Note that $$I_{(0,1)}(tu)\cdot I_{(0,1)}(t-tu)=I_{(0,2)}(t)\cdot I_{(0,1)}(u)\cdot I_{(1-1/t,1/t)}(u), $$ hence $$ f_{T,U}(t,u)=|t|\cdot\left(I_{(0,1)}(t)\cdot I_{(0,1)}(u)+I_{(1,2)}(t)\cdot I_{(1-1/t,1/t)}(u)\right). $$ Integrating this with respect to $u$ yields $$ f_T(t)=|t|\cdot\left(I_{(0,1)}(t)+(2/t-1)\cdot I_{(1,2)}(t)\right), $$ thus, $$ f_T(t)=t\cdot I_{(0,1)}(t)+(2-t)\cdot I_{(1,2)}(t). $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.