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.

  • $\begingroup$ Are you looking for a PDF of the sum of these variables? If so, you'll want some sort of gamma distribution $\endgroup$ – 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


where $f_{X_i}$ is the density of $X_i$

Because they are identically distributed , we can write :



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