Here's the set up:
Let the random variable $X$ have the following distribution
$$f(x)=
\begin{cases}
e^{-x} & \quad \text{if}\hspace{2mm} 0<x<\infty\\
0 & \quad \text{elsewhere.}
\end{cases} $$
Let $Y=[X]$ be the integer part and $Z=X-[X]$ be the fractional part.
Problem:
I'm supposed to find the joint distribution function of $Y$ and $Z$, and the moment generating function of $Y$. I think I have the mgf of $Y$, but don't really know how to approach finding the joint distribution.
My inner dialogue:
It seems like to find $\mathbb{P}(Y\leq t)$, we'll be summing $e^{-x}$ in $x$ over the integers up to $\lfloor t \rfloor$. On the other hand, with $\mathbb{P}(Z\leq t)$, will we integrate, just skipping integers? But $\mathbb{P}(Z\leq t)=\mathbb{P}(Z< t)$ if $Z$ is continuous? Hmm…
I would very much appreciate any suggestions on how to get started here.
Also, this isn't homework.
Best Answer
This is not a complete answer, but just to state what $Y, Z$ should be.
If $t$ is an integer, $\mathbb{P}(Y \leq t) = \mathbb{P}( \lfloor X \rfloor \leq t) = \mathbb{P} ( X < t+1) = \int_0^{t+1} e^{-x}\, dx$, not summing at the integer points.
If $0 \leq r \leq 1$, then $\mathbb{P}(Z \leq r) = \mathbb{P}( X - \lfloor X \rfloor < r) = \sum_{n=0}^\infty \mathbb{P} (n < X < n+r)$.