For each PDF family below, show whether or not the family belongs to the exponential class of densities.

(a) \(f(x ; \beta)=\beta x^{-(\beta+1)} I_{(1, \infty)}(x), \beta \in \Omega=(0, \infty)\). (This is a subfamily of the Pareto family of PDFs.)

(b) \(f(x ; \Theta)=(1 /(2 \Theta)) \exp (-|x| / \Theta), \Theta \in \Omega=(0, \infty)\). (This is a subfamily of the double exponential family of PDFs.)

(c) \(f(x ; \mu, \sigma)=(1 /(x \sqrt{2 \pi} \sigma)) \exp \left(-(\ln (x)-\mu)^{2} /\left(2 \sigma^{2}ight)ight)\). \(I_{(0, \infty)}(\mathrm{x}),(\mu, \sigma) \in \Omega=\{(\mu, \sigma): \mu \in(-\infty, \infty), \sigma>0\}\) (This is the log-normal family of PDFs.) Hint: Expanding the square in the exponent of \(e\) may be helpful as an alternative representation of the exponent.

(d) \(f(x ; r)=((1-r) / r) r^{x} I_{\{1,2,3, \ldots\}}(x), r \in \Omega=(0,1)\).