Let $X$ be a random variable whose distribution depends on an unknown parameter $\theta$. Explain what is meant by a sufficient statistic $T(X)$ for $\theta$.

In the case where $X$ is discrete, with probability mass function $f(x \mid \theta)$, explain, with justification, how a sufficient statistic may be found.

Assume now that $X=\left(X_{1}, \ldots, X_{n}\right)$, where $X_{1}, \ldots, X_{n}$ are independent nonnegative random variables with common density function

$$f(x \mid \theta)= \begin{cases}\lambda e^{-\lambda(x-\theta)} & \text { if } x \geqslant \theta \\ 0 & \text { otherwise }\end{cases}$$

Here $\theta \geq 0$ is unknown and $\lambda$ is a known positive parameter. Find a sufficient statistic for $\theta$ and hence obtain an unbiased estimator $\hat{\theta}$ for $\theta$ of variance $(n \lambda)^{-2}$.

[You may use without proof the following facts: for independent exponential random variables $X$ and $Y$, having parameters $\lambda$ and $\mu$ respectively, $X$ has mean $\lambda^{-1}$ and variance $\lambda^{-2}$ and $\min \{X, Y\}$ has exponential distribution of parameter $\lambda+\mu$.]