Finding probability distribution of a random variable from the others

Question

Let $X_1,...,X_n$ be independent exponential distribution.and now i want to calculate the density function of $\sum_{i=1}^{n} X_i$,i tried to find its distributing function,$F(T(X)\le x) $ but then i don't know what to do.

Answer 1

Suppose your random variables are i.i.d. exponential with parameter $\lambda$. Then the sum $\sum X_i$ is a gamma random variable with parameter $(n, \lambda)$. Here's a sketch of the proof: (you should fill in the details)

1. Let $X$ and $Y$ be gamma random variables with parameters $(s,\lambda)$ and $(t, \lambda)$, respectively. (Look up the gamma density function, if necessary).

2. Prove that $X+Y$ is a gamma random variable with parameter $(s+t, \lambda)$.

3. Prove by induction that if $X_1, \dots, X_n$ are gamma$(t_i, \lambda)$ random variables, then $\sum X_i$ is a gamma$(\sum t_i, \lambda)$, random variable.

4. Note that an exponential rv with parameter $\lambda$ is a gamma$(1,\lambda)$ random variable.