The motivation for considering exponential families of distributions

Question

I saw problems of the form: "show that the distributions ... form an the exponential family". Why is this property, being an exponential family, important?

Answer 1

The exponential family of probability distributions share a number of nice properties, so sometimes, in order to show that a particular distribution has such a property, it suffices to demonstrate that it belongs to the exponential family.

For example, the exponential family of distributions have conjugate priors. So, if I give you a probability distribution with density

$$f_X(x) = xe^{-x}, \quad x > 0,$$

then you can prove that this distribution has a conjugate prior because it is a special case of a gamma distribution, whose general form is $$f_X(x) = \frac{b^a x^{a-1} e^{-bx}}{\Gamma(a)}, \quad x > 0,$$ for the choice $a = 2$, $b = 1$. And since the gamma distribution is a member of the exponential family, it has a conjugate prior:

This is just one particular property (existence of a conjugate prior). There are other "nice" properties, such as sufficiency, that are also shared by the exponential family.