#"We take a time period with length "t", consisting out of n pieces"#
#Delta t = t/n". Suppose that the chance for a successful event"#
#"in one piece is "p", then the total number of events in the n"#
#"time pieces is distributed binomial according to"#
#p_x(x) = C(n,x) p^x (1-p)^(n-x) , x = 0,1,...,n#
#"with "C(n,k) = (n!)/((n-k)!*(k!))" (combinations)"#
#"Now we let"#
#n->oo", so " p->0, " but "n*p = lambda#
#"So we substitute "p=lambda/n" in "p_x" : "#
#p_x(x) = (n!)/((x!)(n-x)!)(lambda/n)^x(1-lambda/n)^(n-x)#
#= lambda^x/(x!)(1-lambda/n)^n(n!)/((n-x)!)*1/(n^x (1-lambda/n)^x)#
#= lambda^x/(x!)(1-lambda/n)^n[(n(n-1)(n-2)...(n-x+1))/(n(1-lambda/n))^x]#
#"for "n -> oo" what is in between [...]" -> 1" and"#
#(1 - lambda/n)^n -> e^-lambda " (Euler's limit),"#
#"so we obtain"#
#p_x(x) = (lambda^x e^-lambda) / (x!) , x = 0,1,2,..., oo#
