Let $Z$ be an exponential random variable with parameter $\lambda=1$. Show that

$$\mathbb{P}(Z>s+t \mid Z>s)=\mathbb{P}(Z>t)$$

for any $s, t \geqslant 0$.

Let $Z_{\text {int }}=\lfloor Z\rfloor$ be the greatest integer less than or equal to $Z$. What is the probability mass function of $Z_{\text {int }}$ ? Show that $\mathbb{E}\left(Z_{\text {int }}\right)=\frac{1}{e-1}$.

Let $Z_{\mathrm{frac}}=Z-Z_{\mathrm{int}}$ be the fractional part of $Z$. What is the density of $Z_{\mathrm{frac}}$ ?

Show that $Z_{\text {int }}$ and $Z_{\text {frac }}$ are independent.