Exercise 1. Let $X_{1}, \cdots, X_{n}$ be i.i.d. RVs with the following PDF of exponential form $$ f_{X ; \theta} (x)=\exp (Kx) p(\theta)+(x)+(\theta)) \mathbf{1} (x \in A), $$ where the support $A$ does not depend on the parameter $\theta$. (i) Show that the joint likelihood function is given by $$ f\left(x_{1}, \cdots, x_{n} ; \theta ight)=\exp \left(p(\theta)\left(\sum_{i=1}^{n} K\left(x_{i} ight) ight)+(\theta) ight) \exp \left(\sum_{i=1}^{n} S\left(x_{i} ight) ight) \mathbf{1}\left(x_{1}, \cdots, x_{n} \in A ight). $$ Deduce that $\sum_{i=1}^{n} K\left(X_{i} ight)$ is a sufficient statistic for $\theta$. (ii) Derive sufficient statistics for the parameters of $\operatorname[Bernoulli}(p), operatorname{Exp}(\lambda)$, and Poisson $(\lambda$ using (i). SP.PB. 086