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## Main Question or Discussion Point

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.

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.

Useful information:

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 ∞).

Any help would be much appreciated! : )

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.

Useful information:

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 ∞).

Any help would be much appreciated! : )