Practice Homework

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

\[ f_Y(y) = a(y,\phi)\exp\left\{\frac{y\theta - b(\theta)}{\phi}\right\} \]

• \(\theta\): canonical parameter

• \(b(\theta)\): cumulant function

• \(\phi>0\): dispersion parameter

• \(a(y,\phi)\): normalizing function

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

• Normal

• Poisson

• Binomial

• Gamma