# How can I find the joint PDF for $n$ independent exponential variables?

I'd like to find a maximum likelihood estimator for $n$ iid exponential random variables. For that I need the joint PDF for those $n$ variables. How can I compute that? In one dimension I would try to work out the cumulative distribution function and then differentiate, but in multiple variables I don't even know where to begin.

• Are you looking for a PDF of the sum of these variables? If so, you'll want some sort of gamma distribution – Omnomnomnom Dec 28 '16 at 14:25

In this situation, the variables $X_i$ are independent, therefore the density function of $(X_1,...,X_n)$ is defined as
$$f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1}(x_1)*...*f_{X_n}(x_n)$$
where $f_{X_i}$ is the density of $X_i$
$$f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1}(x_1)*...*f_{X_1}(x_n)$$