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

con list=3358 el n mero total de pasos es

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

por lo que con list se tiene la distribuci n binomial

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