Show that $Y$ and $Z$ are independent and find their distributions Suppose that $X\sim\exp(\lambda=1)$. Let $Y$ be the integer part and $Z$ be the fractional part. Show that $Y$ and $Z$ are independent and find their distributions. 
This one is kinda confusing. Any help?

For $n\in\mathbb{N},n\geq 0$, we have
$$
\Pr(Y\leq n)=\Pr(X< n+1)=1-e^{-n-1}.
$$
For $z\in[0,1)$, we have
\begin{align*}
\Pr(Z\leq z)&=\sum_{m=0}^\infty\Pr(X\in[m,m+z])\\
&=\sum_{m=0}^\infty(\exp(-m)-\exp(-m-z))\\
&=(1-e^{-z})\sum_{m=0}^\infty(1/e)^m\\
&=\frac{e-e^{1-z}}{e-1}
\end{align*}
Finally,
\begin{align*}
\Pr(Y\leq n,Z\leq z)&=\sum_{m=0}^n\Pr(m\leq X\leq m+z)\\
&=\sum_{m=0}^n(\exp(-m)-\exp(-m-z))\\
&=(1-e^{-z})\sum_{m=0}^n(1/e)^m
\end{align*}
which you can verify to be equal to $\Pr(Y\leq n)\Pr(Z\leq z)$.
