Distribution of Sum of Random Variables

Question

I have joint Cumulative Distribution Function (CDF) of a group of Continuous Random Variables $\{X_1, X_2, ..., X_n\}$ as $F\left(x_1, x_2,..., x_n\right)$.

Having just this information, is there any possible to way to get the Cumulative Distribution Function of the Random Variable $Y = \sum_{i=1}^{n} X_i$.

Any pointer will be very helpful.

Answer 1

By definition,

$$F(x_1,\ldots,x_n) = \Pr(X_1\le x_1,\ldots,X_n\le x_n).$$

Assuming these variables are absolutely continuous means there is a density function $f$ for which

$$F(x_1,\ldots, x_n) = \int^{x_n}\cdots\int^{x_1} f(t_1,\ldots, t_n)\,\mathrm{d}t_1\cdots\mathrm{d}t_n.$$

Observe two things:

1. The Fundamental Theorem of Calculus, applied $n$ times to the right hand side, yields $$f(x_1,\ldots,x_n) = \frac{\partial^n F}{\partial x_1\cdots \partial x_n}(x_1,\ldots, x_n).$$

2. For any number $y,$ the event $Y\le y$ is the event $X_1+\cdots+X_n\le y$ whose probability (therefore) is given by $$\Pr(Y\le y) = \Pr(X_1+\cdots+X_n\le y) = \int\cdots\int_{x_1+\cdots+x_n\le y} f(t_1,\ldots,t_n)\,\mathrm{d}t_1\cdots\mathrm{d}t_n.$$

These assembled observations produce the formula

$$ F_Y(y) = \Pr(Y\le y) = \int\cdots\int_{x_1+\cdots+x_n\le y} \frac{\partial^n F}{\partial t_1\cdots \partial t_n}(t_1,\ldots, t_n)\,\mathrm{d}t_1\cdots\mathrm{d}t_n.$$

That's as far as you can go at this broad level of generality.