Does the binomial distribution belong to the linear exponential family?
I understand that if a random variable $X$ has the density function $$f(x)=\frac{p(x)e^{r(\theta)x}}{q(\theta)},$$ then the distribution of $X$ belongs to the linear exponential family. How can I apply this to prove whether or not the binomial distribution belongs to the linear exponential family? Thanks for the help!
Yes it does. If $\theta=P(S)$ (the probability of success) then you can rewrite the pmf of a binomial by $p_X(x)=C(n,x)\theta^x(1-\theta)^{n-x}$=$\frac{n!}{x!(n-x)!}(\frac{\theta}{1-\theta})^x(1-\theta)^n$=$\frac{n!}{x!(n-x)!}e^{(ln(\frac{\theta}{1-\theta}))x}(1-\theta)^n$. So this fits the form of the exponential family where $r(\theta)=\ln\left(\theta/(1-\theta)\right)$.