Talk:Monte Carlo methods

Python shell sessions as examples

I noted that someone had changed another Python shell session used as an example, into the 'normal' definition of a function followed by the shell session just used to show the answer when the function is called.
I don't think this should be done here, as I am attempting to show how the shell might be used for such a task. It is still Python. The repitition of the input expression is because in idle, the built-in graphical IDE for Python, you would hit return in a previous expression to re-enter it. In the non-graphical shell, you can scroll through previous input to re-enter lines. It can give the immediate feedback, and 'spirit of exploration' you get when working with a calculator. --Paddy3118 05:22, 2 October 2008 (UTC)

Error formula in C implementation

What formula is being used for the error calculation in the C Implementation?

At first I thought it was the formula for standard deviation but the code is:

error = val * sqrt(val (1 - val) / sampled) * 4;

The factor 4 is explained because we are not interested in the ratio $\pi /4$, but in $\pi$ so both the value and the error must be multiplied by 4. The rest of the code translates to:

$\sigma =\mu {\sqrt {{\frac {1}{N}}\mu (1-\mu )}},{\rm {\ \ where\ \ }}\mu {\rm {\ \ is\ \ the\ \ ratio\ \ }}\pi /4{\rm {\ \ and\ \ }}N{\rm {\ \ is\ \ the\ \ number\ \ of\ \ samples\ \ }}$

But according to Wikipedia the formula is this:

$\sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\rm {\ \ where\ \ }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.$

Can somebody explain this more clearly? I'm not yet convinced this is correct

Randomly throwing a point in a square, and it has chance p of being in the circle, while (1-p) chance of otherwise. If you throw N points and count the number of times n that they landed in the circle, n would follow wp:binomial distribution (look up the variance formula there). Here we are taking p = n/N as the ratio between areas of circle and square, but n is subject to statistical fluctuation. Assuming that we had the senses to throw a large enough N so n/N wouldn't be a completely bogus estimate of p, but we'd still like to know how far off it could be from p's true value. This is where the variance comes in: it tells you, given N and a rough knowledge of p, how much uncertainty of n (and p) one should expect.

If you want to use the stddev formula, then each $x_{i}$ takes the value of either 1 (landing in circle) or 0 (not). The average is $\mu =p$ as mentioned above; now $\sigma ^{2}={1 \over N}\sum (x_{i}-p)^{2}$. Note that there are going to be about $Np$ of those $x_{i}$s with value $1$, and $N(1-p)$ with value $0$, so $\sum (x_{i}-p)^{2}\approx Np(1-p)^{2}+N(1-p)(0-p)^{2}=Np(1-p)$. See how it comes back to the same formula? --Ledrug (talk) 06:17, 5 May 2014 (UTC)

Thank you very much that explains the origin of the formula. Even though following your reasoning the formula should be:

$\sum (x_{i}-p)^{2}\approx Np(1-p)$,

But because we have a factor of ${1N}$ we must take into account, then the resulting stddev should be:

$\sigma ^{2}=p(1-p)$,

Meaning that the formula implemented has an extra factor of ${1 \over N}$ inside the square root and a factor of $p$ outside of the square root, meaning:

error = val * sqrt(val * (1 - val) / sampled) * 4, when it should be:

error = sqrt(val * (1 - val)) * 4;

Am I missing something else here? Sorry for the intrigue I'm no expert in probability, but I'm curious as to the implementation.-Chibby0ne (talk) 17:18, 17 May 2014 (UTC)