Suppose $X$ is a Gaussian random variable, i.e., $X\sim N(\mu, \sigma)$. Let $Y$ be defined as $Y=\lfloor X \rfloor$, where $\lfloor \cdot \rfloor$ denotes the floor function (greatest integer lesser or equal than $X$). It can be seen from here that $$P(Y=y)=P(y\le X< y+1)=P(X<y+1)-P(X\le y)\\ =\Phi\left(\frac{y+1-\mu}{\sigma}\right)-\Phi\left(\frac{y-\mu}{\sigma}\right) \\ =\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{y+1-\mu}{\sigma\sqrt{2}}\right)\right]-\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{y-\mu}{\sigma\sqrt{2}}\right)\right]$$ where $\Phi(\cdot$) is the CDF for standard Gaussian and $\operatorname{erf}(\cdot)$ is the error function.
I have two questions:
(1) Is $Y$ a regular discrete random variable? Or it does not have a standard expression (i.e., does it belong to any families of distributions)?
(2) More importantly, what is the mean and variance of random variable $Y$?
Please help and thanks in advance!
Best Answer
Note that you can write $X = \lfloor X \rfloor + \{X\}$, where $\{X\}$ is the fractional part of $X$. Using the Poisson summation formula, I get (if I'm not mistaken)
$$ \mathbb E\lfloor X \rfloor = \mu - \mathbb E \{X\} = \mu - \frac{1}{2} + \frac{1}{\pi} \sum_{j=1}^\infty e^{-2\pi^2\sigma^2 j^2} \frac{\sin(2\pi j \mu)}{j} $$