- Let $X_1,X_2$ be independent random variables each having a exponential distribution with mean $λ = 1$.
(a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$.
(b) Get the marginal density of $f_{Y_1}(y_1)$ and $f_{Y_2}(y_2)$.
These are what I've found so far.
$$f_{X_1}(x_1)=e^{-x_1}, f_{X_2}(x_2)=e^{-x_2}$$ $$f_{X_1,X_2}(x_1, x_2)=f_{X_1}(x_1)f_{X_2}(x_2)=e^{-x_1-x_2}$$
Also, since $Y_1=X1$, $f_{Y_1}(y_1)=e^{-y_1}$ right? Similarly, since $Y_2=X_1+X_2$, $f_{Y_2}(y_2)=e^{-y_2}$
However, is it guaranteed that $Y_1$ and $Y_2$ are independent? or else how can I solve these problems??
Oh, and I haven't learned the Jacobian so it would be grateful if you avoid using it.