Let $X$ be a sample (size n = 1) from the exponential distribution, which has the pdf $$f(x;\lambda) = \lambda \exp(-\lambda x)$$ where $\lambda$ is an unknown parameter. Let's define a statistic as
$$T(X)= \left\{ \begin{array}{lcc} 1 & X > 2 \\ \\ 0 &\mbox{ otherwise} \\ \\ \end{array} \right.$$
Is $T(X)$ a sufficient statistic for $\lambda$?
I honestly have no idea on how to approach this problem. Any suggestions?
Use the definition of sufficiency to prove that a statistic is not sufficient; the Factorisation theorem comes in handy when you are proving that a statistic is sufficient.
For any $x\geqslant 0$, the distribution function of $X\mid T=1$ is
\begin{align} P(X\leqslant x\mid T=1)&=\begin{cases}\frac{P(X\leqslant x)}{P(T=1)}&,\text{ if }T=1 \\ \quad 0 &,\text{ otherwise }\end{cases} \\&=\begin{cases}\frac{P(X\leqslant x)}{P(X>2)}&,\text{ if }T=1 \\ \quad 0&,\text{ otherwise }\end{cases} \end{align}
Can you see now whether the conditional distribution $X\mid T$ depends on $\lambda$ or not? From this information, conclude about the sufficiency of $T$.