User:


Gaussian distribution
Storyboard

In the limit of similar probabilities the binomial distribution is reduced in the continuous limit to the Gaussean distribution.

>Model

ID:(1556, 0)


Distribución binomial
Equation

>Top, >Model


Con la probabilidad de que se de un numero definido de pasos a la derecha e izquierda esta dada por

$W_N(n_1,n_2)=\displaystyle\frac{N!}{n_1!n_2!}p^{n_1}q^{n_2}$



con el número total de pasos es

$N=n_1+n_2$



y solo existe la probabilidad de ir a la derecha o a la izquierda, con se tiene para las probabilidades que

$p+q=1$



por lo que con se tiene la distribución binomial

$ W_N(n) =\displaystyle\frac{ N !}{ n !( N - n )!} p ^ n (1- p )^{ N - n }$

ID:(8961, 0)


Approach for $N!$
Equation

>Top, >Model


With the Stirling approximation

equation=8966

and the change of variables

equation=8996

you get that

equation

ID:(8998, 0)


Approximation for $n!$
Equation

>Top, >Model


With the Stirling approximation

equation=8966

and the change of variables

equation=11431

you get that

equation

ID:(9003, 0)


Approximation for $(N-n)!$
Equation

>Top, >Model


With the Stirling approximation

equation=8966

and the change of variables

equation=8997

the expression is

equation

ID:(8999, 0)


Factor $N!/N!(N-n)!$ For $N\gg 1$, $n\gg 1$ and $N>n$
Equation

>Top, >Model


In the case of medium probabilities (p \sim q \sim 1/2) and large numbers N it can be shown with

$N!\sim\sqrt{2\pi N}\left(\displaystyle\frac{N}{e}\right)^N$



$n!\sim\sqrt{2\pi n}\left(\displaystyle\frac{n}{e}\right)^n$



and

$(N-n)!\sim\sqrt{2\pi(N-n)}\left(\displaystyle\frac{N-n}{e}\right)^{N-n}$



is obtained

$\displaystyle\frac{N!}{n!(N-n)!}\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}$

ID:(507, 0)


Limit of large Numbers and middle Probabilities
Equation

>Top, >Model


The expression

$ W_N(n) =\displaystyle\frac{ N !}{ n !( N - n )!} p ^ n (1- p )^{ N - n }$



is reduced by

$\displaystyle\frac{N!}{n!(N-n)!}\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}$



to representation

$W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}p^n(1-p)^{N-n}$

ID:(506, 0)


Average position
Equation

>Top, >Model


If total N steps are taken with a probability p in the right direction and these have a length a the expected final position will be

$\mu=aNp$

ID:(9008, 0)


Change variables by offset $x=(n-Np)a$
Equation

>Top, >Model


To obtain the Gaussian distribution it is necessary to develop the distribution around its deviation from its mean position that can be given by

$x=(n-Np)a$

ID:(8973, 0)


Factor $n/N$ depending on the path $x$
Equation

>Top, >Model


As the way is

$x=(n-Np)a$



factor n/N can be written as

$\displaystyle\frac{n}{N}=p\left(1+\displaystyle\frac{x}{aNp}\right)$

ID:(9004, 0)


Factor $N-n/N$ depending on the path $x$
Equation

>Top, >Model


As the way is

$x=(n-Np)a$



factor n/N can be written as

$\displaystyle\frac{N-n}{N}=(1-p)\left(1-\displaystyle\frac{x}{aN(1-p)}\right)$

ID:(9005, 0)


Binomial distribution as a function of deviation
Equation

>Top, >Model


If large numbers and probabilities around 1/2 are entered in the binomial distribution for the case

$W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}}\left(\displaystyle\frac{n}{N}\right)^{-n-1/2}\left(\displaystyle\frac{N-n}{N}\right)^{-N+n-1/2}p^n(1-p)^{N-n}$



the expressions

$\displaystyle\frac{n}{N}=p\left(1+\displaystyle\frac{x}{aNp}\right)$



and

$\displaystyle\frac{N-n}{N}=(1-p)\left(1-\displaystyle\frac{x}{aN(1-p)}\right)$




a distribution of the form is obtained

$W_N(n)\sim\displaystyle\frac{1}{\sqrt{2\pi N}p(1-p)}\left(1+\displaystyle\frac{x}{aNp}\right)^{-n-1/2}\left(1-\displaystyle\frac{x}{aN(1-p)}\right)^{-N+n-1/2}$

ID:(8974, 0)


Variable change $u=x/aNp$
Equation

>Top, >Model


To develop the 1+x/aNp factor you can work with the variable change

$u=\displaystyle\frac{x}{aNp}$

ID:(9021, 0)


Factor $1+x/aNp$ for $N\gg 1$ and $p\sim 1/2$
Equation

>Top, >Model


With the approximation

$1+u\sim e^{u-\frac{1}{2}u^2}$



it has to

$\left(1+\displaystyle\frac{x}{aNp}\right)\sim e^{x/aNp-x^2/2a^2N^2p^2}$

ID:(9006, 0)


Variable change $u=x/aN(1-p)$
Equation

>Top, >Model


To develop the factor 1+x/aN(1-p) you can work with the variable change

$u=\displaystyle\frac{x}{aN(1-p)}$

ID:(9022, 0)


Factor $1-x /aN(1-p)$ for $N\gg 1$ and $p\sim 1/2$
Equation

>Top, >Model


With the approximation

$1+u\sim e^{u-\frac{1}{2}u^2}$



it has to

$\left(1-\displaystyle\frac{x}{aN(1-p)}\right)\sim e^{-x/aN(1-p)-x^2/2a^2N^2(1-p)^2}$

ID:(9007, 0)


Probability for large $N$ and middle $p$
Equation

>Top, >Model


It can be shown that for a large number N and probability p neither too small nor too close to 1, the binomial distribution is reduced to a gausseana for the position x=na:

$P(x)=\displaystyle\frac{1}{\sqrt{2\pi N^2p(1-p)}}e^{-(x-aNp)^2/2N^2p(1-p)}$

In this case, the probability q was replaced by 1-p.

ID:(3367, 0)


Generalization Limit Big Numbers
Equation

>Top, >Model


$\begin{matrix}

P(x) & = & \displaystyle\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/2\sigma^2}\\

\sigma^2 & = & Np(1-p)\\

\end{matrix}

$

ID:(3368, 0)


Gaussian distribution standard deviation
Equation

>Top, >Model


The standard deviation of the binomial distribution at the limit N large and p medium is

$ \sigma^2 = N ^2 p (1- p )$

ID:(8963, 0)


Example comparison with Gaussian distribution
Image

>Top


If we study the binomial distribution for large numbers N and probabilities around 1/2

$P(x)=\displaystyle\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/2\sigma^2}$



which is represented below:

ID:(7793, 0)


0
Video

Video: Gaussian distribution