Practice Homework

A random variable $Y$ is can belong to an exponential family of distributions if the density function can take the following form:

$$f_Y(y)=a(y,\varphi )\exp \left\{\frac{y\theta -b(\theta )}{\varphi }\right\}$$

• $\theta$: canonical parameter

• $b(\theta )$: cumulant function

• $\varphi >0$: dispersion parameter

• $a(y,\varphi )$: normalizing function

Convert the following distributions to the exponential form of the probability density function:

• Normal

• Poisson

• Binomial

• Gamma