Is the mixture of Exponential family distributions an Exponential family distribution too? Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. Is the new distribution an exponential family too?
 A: In general the answer is "no".
The mixture distribution
$$
\alpha h_1(x)\exp^{\theta_1^T \phi_1(x) - A_1(\theta_1)} + (1-\alpha) h_2(x)\exp^{\theta_2^T \phi_2(x) - A_2(\theta_2)}
$$
generally can't be factored into something like
$$
h_3(x) \exp^{\theta_3^T \phi_3(x) - A_3(\theta_3)}
$$
even if the two components come from the same exponential family (so $h_1=h_2, \phi_1=\phi_2, A_1=A_2$). A classic example is a mixture of Gaussians.

Although the product of exponential families is an (unnormalized)
  exponential family, the mixture of exponential families is not an
  exponential family.

--page 6 of Statistical exponential families: A digest with flash cards
A: In my viewpoint, this depends on the nature of the mixed distributions. Without loss of generality, consider a probability distribution with a single rate parameter $\theta$. The latter probability distribution belongs to the exponential family if its PDF can be expressed in the form
$$
f(x | \theta) = e^{\eta(\theta)T(x)-A(\theta)+B(x)}
$$
Now, consider an exponential distribution with rate parameter $\theta$ and a gamma distribution with known shape parameter $2$ and rate parameter $\theta$ with corresponding PDFs
$$
f(x | \theta) = \theta e^{-\theta x} \equiv f(x | \theta) =  e^{-\theta x + \log \theta + 0}
$$
and
$$
f(x | \theta) = \theta^2 x e^{-\theta x} \equiv f(x | \theta) =  e^{-\theta x + 2 \log \theta + x}
$$
respectively. By mixing these distribution using the weights $\theta(1 + \theta)^{-1}$ and $(1 + \theta)^{-1}$, respectively, we will get the PDF of the Lindley distribution that is given by
$$
f(x | \theta) = \theta^2 (1 + \theta)^{-1} (1 + x) e^{-\theta x} \equiv f(x | \theta) =  e^{-\theta x +  \log \left\lbrace\theta^2 (1 + \theta)\right\rbrace + \log(1 + x)}
$$
My personal verdict is that there might be "a few" special cases when a mixture of $f$s, where $f$ is a distribution of the exponential family, will yield a distribution that belongs to the exponential family.
