Use the Lehman-Scheff minimal sufficiency theorem, or some other argument, to find a set of minimal sufficient

Use the Lehman-Scheffé minimal sufficiency theorem, or some other argument, to find a set of minimal sufficient statistics for each case below.

a. You are random sampling from a log-normal population distribution given by

\(f\left(z ; \mu, \sigma^{2}ight)=\frac{1}{(2 \pi)^{1 / 2} \sigma z} \exp \left(-\frac{1}{2 \sigma^{2}}(\ln (z)-\mu)^{2}ight) I_{(0, \infty)}(z)\), where \(\mu \in(-\infty, \infty)\) and \(\sigma^{2}>0\).

b. You are random sampling from a "power function" population distribution given by

\(f(z ; \lambda)=\lambda z^{\lambda-1} I_{(0,1)}(z)\), where \(\lambda>0\).

c. You are random sampling from a Poisson population distribution \(f(x ; \lambda)=\frac{e^{-\lambda} \lambda^{x}}{x !} I_{\{0,1,2, \ldots,\}}(x)\).

d. You are random sampling from a negative binomial density \(f\left(x ; r_{0}, pight)=\frac{(x-1) !}{\left(r_{0}-1ight) !\left(x-r_{0}ight) !} p^{r_{0}}(1-p)^{x-r_{0}} I_{\left\{r_{0}, r_{0}+1, \ldotsight\}}(x)\)
where \(r_{0}\) is a known positive integer.

e. You are random sampling from a \(N\left(\mu, \sigma^{2}ight)\) population distribution.

f. You are random sampling from a continuous uniform density \(f(x ; \Theta)=\frac{1}{\Theta} I_{(0, \Theta)}(x)\)
g. You are sampling from a Beta distribution \(f(x ; \alpha, \beta)=\frac{1}{B(\alpha, \beta)} x^{\alpha-1}(1-x)^{\beta-1} I_{(0,1)}(x)\).