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A particle moves along an axis, and every instant it has a 50% chance of going up and a 50% chance of going down. If its speed is 1, after 1 second the particle will be somewhere between -1 and 1. I'd like to find the function $f(x)$ whose integral between $a$ and $b$ gives the probability of finding the particle in that area.

Obviously, $\int_{-1}^1f(x)dx=1$; it looks very similar to a normal distribution, apart from the fact that $f(x)=0$ for $x\le -1$ and $x\ge 1$.

Edit: now I believe it looks more like a dirac delta function, but I'm not completely certain.

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    can you provide a more formal definition of your process? is it a random walk, a Brownian motion...? – 2017-02-08
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    I don't know really know what those things are, but it's probably a random walk? Anyway, I am now almost completely certain that it's a dirac delta function. – 2017-02-08

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This is my reasoning (for convenience, the particle now starts at 0.5 and has a speed of 0.5):

Let's start with a particle that, at the start, decides to either go up or down. In this case, After one second it'll have a 50% chance of being at 0 and a 50% of being at 1.

If, instead, the particle first decide at 0 what direction to have, and then decides again after 0.5s, after 1s it'll have a 25% chance of being at 0, a 50% chance of being at 0.5, and a 25% chance of being at 1.

In general, if the particle decides every $1/n$ s what direction to have, it'll have a probability of landing on $a/n$ of $\frac{\binom{n}{a}}{2^n}$ (for $0\le a \le 1)$. So, let's define $g$ as $g(\frac{a}{n})=\frac{\binom{n}{a}}{2^n}$, so $g(a)=\frac{\binom{n}{an}}{2^n}$.

The problem with $g$ is that $\int_0^1g(x)dx\ne1$; to fix this problem I define a new function $f$ as $f(x)=\frac{g(x)}{\int_0^1g(x)dx}$ (now, $\int_0^1f(x)dx=1$); perhaps it's not a very rigorous step, but I feel like it makes sense.

The original problem is just the special case $\lim_{n\to \infty}$; graphing $f(x)$, it's pretty evident that the function approaches the dirac delta function (here's some graphs:https://www.desmos.com/calculator/nwew2tplqj).

(Does all of this make any sense?)