Distribution of the sum of n independent variables of the exponential family.

Question

Suppose you have $n$ random and independent variables $Y_{1},...,Y_{n}$ whose distribution belongs to the uniparametric exponential family.

How do I find the distribution of $\sum_{i=1}^{n} Yi$ ?

Answer 1

Let $Y_i \sim \exp(1)$, i.e. $f_Y(y) = e^{-y}$ and $F_y(y)=1-e^{-y}$. You can extend the analysis easily to $Y_i \sim \exp(\lambda)$.

Let's consider the case of $n=2$, i.e. $X = Y_1+Y_2$.

$F_X(x) = P(X \leq x)$

= $P(Y_1+Y_2 \leq x)$

= $\int_0^x P(Y_1 \leq x-y) f_Y(y) dy$

= $\int_0^x \left( 1-e^{-(x-y)}\right) e^{-y} dy$

= $\int_0^x (e^{-y} - e^{-x})dy$

= $1-e^{-x}-xe^{-x}$

You can differentiate to get the PDF $f_X(x) = xe^{-x}$.

If you do this a couple more times, you will see a pattern, at which point you can arrive at the answer by the principle of mathematical induction. The moment generating function approach is of course much quicker, if you are familiar with that.