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Sums of exponentially distributed rvs

Hi,

Can anyone derive the sum of exponentially distributed random variables?

I have the derivation, but I'm confused about a number of steps in the derivation.

Here they are:

Random variable x has the PDF,

$$f(s) = \left\{ \begin{array}{c l} e^{-s} & if s \ge 0 \\ 0 & otherwise \end{array} \right.$$

Let $$X_1, X_2, .... , X_n$$ be independently exponentially distributed random variables.

The PDF of the sum, $$X_1 + X_2 + ..... +X_n$$ is

$$q(s) = e^{-(s_1+s_2+....+s_n)}$$ where s $$s \ge 0$$

=> $$\int_{a \le s_1+s_2+....+s_n \le b} q(s) ds$$

= $$\int_{a \le s_1+s_2+....+s_n \le b} e^{-(s_1 + .... + s_n)} ds$$

Can anyone explain this stage? Going from the above integral to the following integral?

= $$\int^b_a e^{-u} vol_{n-1} T_u du$$

where $$T_u = [s_1+ .... + s_n = u]$$

What would $$vol_{n-1}$$ be here?

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