Gaussian distributions for the toss of a coin I toss a coin, if i get head i move for a +1 step, if i get tail i move for a -1 step.
I toss this coin N times (first N=10000, then N=40000,then N=160000), and i define a random variable D, as the distance traveled.
Is is possible to obtain these probability density functions for these 3 experiment ? What i thought is that : growing the number N, the gaussian should have become tighter, not the contrary...

 A: A random variable which can take the values $+1$ or $-1$ with equal probability is called a Rademacher distribution.  
It has mean $0$, variance $1$ and standard deviation $1$.
Take a sample size $n$ and adding them up gives a random variable with mean $0$, variance $n$ and standard deviation $\sqrt{n}$.  This is what you are seeing: 


*

*with $n=10000$ you get a standard deviation of $100$

*with $n=40000$ you get a standard deviation of $200$

*with $n=160000$ you get a standard deviation of $400$


which is increasing as $n$ increases. In the large majority of cases you will see a result within two standard deviations of the mean; the Central Limit Theorem suggest an approximately Gaussian distribution for large $n$.
If instead of summing, you had taken the average of the $n$ samples, you a random variable with mean $0$, variance $\frac1n$ and standard deviation $\frac1{\sqrt{n}}$:


*

*with $n=10000$ you would get a standard deviation of $\frac1{100}$

*with $n=40000$ you would get a standard deviation of $\frac1{200}$

*with $n=160000$ you would get a standard deviation of $\frac1{400}$


and this would have led to the distribution getting "tighter" as $n$ increases.
