Finding the Gamma Distribution of X1+...+Xn

[glacier302]

Homework Statement

Let X1,...,Xn be independent, identically distributed random variables with exponential distribution of parameter λ. Find the density function of S = X1+...+Xn. (This distribution is called the gamma distribution of parameters n and λ). Hint: Proceed by induction.

Homework Equations

The probability density function of each of the Xi is f(x) = λe^(-λx).

The probability density function of the sum of two independent random variables is the convolution of their density functions. So if the density function of X is f(x) and the density function of Y is g(x), then the density function of X+Y is ∫f(T)g(x-T)dT (integral from -∞ to ∞).

The Attempt at a Solution

At first I tried computing the characteristic function of X1+...+Xn, which is equal to the characteristic function of X1 raised to the nth power since the Xi are independent and identically distributed. But this didn't look like the characteristic function of any probability distribution that I know, so that was a dead end.

We're told to proceed by induction, but I'm not sure how to do that with density functions.

Any help would be much appreciated! : )

 

[LCKurtz]

I would begin by actually calculating that convolution integral. So letting f(x) represent your exponential distribution λe^-λx you want to calculate

$$\int_{-\infty}^\infty \int_{-\infty}^\infty f(t)f(x-t)\, dt$$

But you need to take care about the limits. Remember the exponential distribution is zero whenever its argument is negative. So your first problem is to put in the exponential functions with the correct limits for the non-zero domain. Then just do the integral and look at its form. Then you will be ready to try the induction step.