I'm new here. I'm trying to proof that the exponential distribution belongs to the exponential family, but I don't know how to do that. Can you help me? Thanks a lot.

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    $\begingroup$ There’s evidence on the Wikipedia article for exponential family distributions. This sounds like a homework/practice question though, so please tag as self-study. What have you tried so far? What do you already understand, and where are you stuck. $\endgroup$ – Arya McCarthy May 5 at 17:59


From Wikipedia:

A single-parameter exponential family is a set of probability distributions whose probability density function (or probability mass function, for the case of a discrete distribution) can be expressed in the form $f_X(x\mid\theta) = h(x)\,\exp\!\bigl[\,\eta(\theta) \cdot T(x) +A(\theta)\,\bigr]$ where $T(x),h(x),η(θ),$ and $A(θ)$ are known functions.


The probability density function (pdf) of an exponential distribution is $$ f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}$$

Your mission, should you choose to accept it, is to find suitable $T(x),h(x),η(θ),$ and $A(θ)$ to demonstrate the exponential distribution pdf is of an exponential family form. It is not particularly difficult, especially if you spot $\exp\!\bigl[\,\eta(\theta) \cdot T(x) +A(\theta)\,\bigr] = e^{A(\theta)}e^{\eta(\theta) \cdot T(x)}$. You can presume $\lambda=\theta$.


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