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Poisson distributions

Storyboard

In the case where the probability is very small, the binomial distribution is reduced to a Poisson distribution.

>Model

ID:(1555, 0)



Example comparison with Poisson distribution

Definition

If we study the binomial distribution for large numbers N and very small probability p \ ll 1 it can be approximated using a Poisson distribution. The comparison can be done with the following simulator:

ID:(7794, 0)



Poisson distributions

Storyboard

In the case where the probability is very small, the binomial distribution is reduced to a Poisson distribution.

Variables

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
e^{-\lambda}
elam
Exponential e^{-\lambda}
-
N^n
N^n
Exponential N^n
-
n!
n!
Factorial n!
-
n
n
Number
-
N
N
Número total de pasos
-
n
n
Número totales de pasos a la izquierda
-
\lambda^n
lambda_n
Power of lambda \lambda^n
-
P_N(m)
P_Nm
Probabilidad de n_1 de N pasos hacia la izquierda
-
p
p
Probabilidad de pasos hacia la izquierda
-
\lambda
lam
Standard Deviation Poisson
-

Calculations


First, select the equation:   to ,  then, select the variable:   to 
P_lambda(n) =( lambda ^ n / n! )*exp(- lambda )N^n\sim\displaystyle\frac{N!}{(N-n)!} W_N(n) =math.factorial( N )* p ^ n *(1- p )^( N - n )/(math.factorial( n )*math.factorial( N - n ))lambda=Npe^{-\lambda}\sim (1-p)^{N-n}\displaystyle\frac{N!}{(N-n)!}p^n\sim \lambda^nelamN^nn!nNnlambda_nP_Nmplam

Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used
P_lambda(n) =( lambda ^ n / n! )*exp(- lambda )N^n\sim\displaystyle\frac{N!}{(N-n)!} W_N(n) =math.factorial( N )* p ^ n *(1- p )^( N - n )/(math.factorial( n )*math.factorial( N - n ))lambda=Npe^{-\lambda}\sim (1-p)^{N-n}\displaystyle\frac{N!}{(N-n)!}p^n\sim \lambda^nelamN^nn!nNnlambda_nP_Nmplam



Equations


Examples

Con list=8970 la probabilidad de que se de un numero definido de pasos a la derecha e izquierda esta dada por

equation=8970



con list=3358 el n mero total de pasos es

equation=3358



y solo existe la probabilidad de ir a la derecha o a la izquierda, con list=8965 se tiene para las probabilidades que

equation=8965



por lo que con list se tiene la distribuci n binomial

equation

Therefore expressions such as N!/(Nn)! for N large (N\gg 1) and n small (N\gg n) can be approximated with

equation=8966

with what you get with N\gg n

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

that is

equation

With the approximation

equation=4738

and employing

equation=8964

it can be shown that

equation

How the exponential is defined as

equation=8967

and by entering

equation=8964

you can replace z=-\lambda=-Np and u=N-n with N\gg n what results

equation

Since the probability of taking n steps in one direction is

equation=8961

for a large number N and the probability is very small p \ll 1 can be approximated

equation=8969

and

equation=8968

the binomial distribution is reduced to a Poisson distribution:

equation

If we study the binomial distribution for large numbers N and very small probability p \ ll 1 it can be approximated using a Poisson distribution. The comparison can be done with the following simulator:

image


>Model

ID:(1555, 0)