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Suppose that a non-negative random variable $X$ has a distribution function $F(x)$, and that $Y$ is the rounding error if $X$ is rounded off to the nearest integer below. Show that $Y$ has the distribution function

$$\sum_{j=0}^\infty [F(j+y) - F(j)] $$ where $$ (0 \le y < 1)$$

My thoughts:

$Y = X- \lfloor X\rfloor$

$P(Y \le y) = P(X-\lfloor X\rfloor \le y)$

So I need to find the density function of $X-\lfloor X\rfloor$.

I first look the density function of $\lfloor X\rfloor$.

$P(\lfloor X\rfloor \le k) = P(0 \le X < k+1)$ where $k$ is an non-zero integer.

$P(0 \le X < k+1) = \int^{k+1}_0f(x)dx = $. Then I don't how should I continue;

ElleryL
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  • @LeeDavidChungLin yes, sorry for the typo – ElleryL Dec 03 '16 at 06:23
  • This is direct, rigorously speaking, using the disjoint union $${Y\leqslant y}=\bigcup_{j=0}^\infty{j\leqslant X\leqslant j+y}$$ for every $0\leqslant y<1$. By the way, the correct formula is rather $$F_Y(y)=\sum_{j=0}^\infty F_X(j+y)-F_X(j^-)$$ where, for every $x$, $$F_X(x^-)=\lim_{t\to x,t<x}F_X(t)=P(X<x)$$ – Did Dec 07 '16 at 10:09

1 Answers1

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$$\begin{align} P(Y \le y) &= P(X-\lfloor X\rfloor \le y) \\ &= P\left(~~ X - 0 \le y \quad\text{or}\quad X - 1 \le y \quad\text{or}\quad X - 2 \le y ~\ldots ~~\right) \\ &= \left[ F(0+y) - F(0)\right] + \left[ F(1+y) - F(1)\right] + \left[ F(2+y) - F(2)\right]+ \ldots \end{align}$$ the probability can be split like that and added since $X$ is non-negative and the cases are mutually exclusive.

Personally I take the above as just a reasonable and intuitive way to "look at" and "understand" the desired result but not a rigorous derivation. This $\sum_{j=0}^\infty [F(j+y) - F(j)]$ is easy to visualize: just cut the density of $X$ at the integers and shuffle/align the curve segments. Note that it indeed starts at $P = 0$ at $y=0$ and goes up to $P \to 1$ as $y \to 1^{-}$.

I've been waiting for someone to give a formal proof, e.g. following the OP's line of thought treating it as $Y = X + W$ where $W \equiv -\lfloor X\rfloor$ and doing the convolution or MGF after finding out the density of $W$ (which is essentially a discrete pmf). This route seems technically tedious and might NOT be illuminating at all, but I believe it is possible and can be useful for more complicated situations.

Anyway, here's my two cents and I hope this question stays visible for longer and someone else more capable can contribute.

Lee David Chung Lin
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  • What makes you think that this approach isn't formal enough? – Hugh Dec 05 '16 at 01:56
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    @Hugh oh it's just that the situation here is simple so to me this is first and foremost "intuitive" than "formal". I'm not implying that if someone else post this as an answer I'm gonna challenge it. It might be useful to have a less intuitive, more technical, and more formal procedure to do this when the core of the situation is less obvious. – Lee David Chung Lin Dec 05 '16 at 02:02