Fractional part of normally distributed variable Let $X$ be a normally distributed variable with mean $0$ and standard deviation $1$. I will consider its fractional part $$\overline{X} = X - \lfloor X \rfloor = X \, \bmod \, 1.$$
I have done some numerical testing and it seems likely that $\overline{X}$ is uniformly distributed on $[0,1].$ To be specific I computed $$\sum_{k=-200}^{200} \Big( \Phi(k+b) - \Phi(k+a) \Big)$$ for a few values of $b \ge a$ in $[0,1]$ and the result is consistently very close to $b-a$.
Here is the code in Sage: I first define
def Phi(x):
    return (1/2 + erf(x / sqrt(2)) / 2).n()

then a few examples of these computations:
s = 0
for k in range(-200,200):
    s = s + Phi(k+3/5) - Phi(k + 2/5)
print s.n()
0.199999998998919

and
s=0
for k in range(-200,200):
    s = s + Phi(k+4/9) - Phi(k + 2/9)
print s.n()
0.222222221674844

Question: is $\overline{X}$ in fact uniformly distributed? From the examples I've done I am confident that it is but I am not sure how to prove it.
 A: $\overline{X}$ is not actually uniformly distributed, but it is ridiculously close to being so.
After some numerical experiments that also convinced me, I tried looking for sources online to confirm this, and found this article. Figure 1 on page 3 plots the PDF of $\overline{X}$ from 0 to 1; it is very close to uniform, but apparently it actually varies between 1.000000004 and 0.999999996.
To be honest, even now I half suspect there's some mistake in the article, because coincidences like this don't just happen. But there's a proof and everything.
There's also an intuitive explanation. To quote the article:

A better explanation relates to a well-known quick and dirty way of
  generating normal variates on a computer by simply summing 12 uniforms
  and subtracting 6. Since the variance of a uniform is 1/12, by the
  central limit theorem, this procedure should have a density that is
  very close to a standard normal. But the distribution of the
  fractional part of the sum of any number of uniforms is exactly
  uniform. The conclusion can only be that the fractional part of a
  standard normal must be very close to uniform.

