I have a conceptual understanding problem. If we have say random variables $X_1$ and $X_2$ and they are i.i.d. with PDF $f$, and we want to find the distribution of the sum of these variables say, $Y$. Why:
- The formula is what I will show, and how does one find a distribution of two random variables combined in general? Or for that matter, n-random variables.
- Why is the second function a function of $y-x_1$? This is my conceptual hurdle.
$$f_Y(y)=\iint f(x_1)f(x_2)\delta (x_1+x_2-y) dx_1 dx_2=\int f(x_1) f(y-x_1) dx_1$$