Here $Y$ is uniform, so $X\mid Y=1$ is a unit exponential

The joint density of X and Y is given by

$$f(x,y)=\begin{cases}\dfrac y2e^{-xy}, & 0<x<\infty, 0<y<2\ 0, & \text{otherwise.}\end{cases}$$

Consider the statements: (I) the conditional density of X given $Y=1$ is $f(x\mid1)=\tfrac12e^{-x}$, $0<x<\infty$ (0 otherwise); (II) $E[X\mid Y=1]=1$; (III) $P[X\le2\mid Y=1]=1-e^{-2}$. Which of the above statements are true?