Suppose that $a,b $ are real constants. That $X_1, X_2$ are independent real random variables with densities $\phi_1, \phi_2$ with respect to the Lebesgue measure. I know that a random variable $Y$ satisfies both $Y = X_1 + a$ as well as $Y = b - X_2$. How would I go about calculating the density of $Y$?
My effort: I know that if I only knew the formula $Y = X_1 + a$ then the density would just be $\phi_1( \cdot - a)$ and similarly if I only had the second equation. I came to this question from the following: If I had a sequence of i.i.d. random variables $X_1, X_2, \dots$ and I defined a new sequence of $Y_i = \sum_{j=1}^i X_j$ for each natural number $i$. Then for some $n$ I look at the conditional distribution of $Y_n$ given the value of all the other $\{Y_i\}_{i \neq n}$ (one may want to write this as $\mathbb{E}(Y_n \mid \sigma(\{Y_i\}_{i \neq n})$). Then I want to prove bounds on the infinity norm on $$\mathbb{E}(Y_n \mid \sigma(\{Y_i\}_{i \neq n}).$$ To do that I notice that all information that you have from knowing $\sigma(\{Y_i\}_{i \neq n})$ amounts to knowing just $Y_{n+1}$ and $Y_{n-1}$. To obtain the question we have that $Y_n = Y_{n-1} + X_n$ and $Y_n = Y_{n+1} - X_{n+1}$, where $Y_{n-1} = a$ and $Y_{n+1} = b$ are know constants. If $X_i$ has density $\phi$ with respect to the Lebesgue measure then I can calculate the distributions of $Y_i$ by convolution $\phi * \phi \dots * \phi$, but it was not apparent to me how to find the joint distribution of say $X_1$ and $X_1+X_2$ with this framework and I thought that posing the above questions might help illuminate what the real problem is.