How can you prove the Poisson Distribution?

Question

Andrew Adams · Accepted Answer

#"See explanation"# #"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#

Quinn Allen · Answer

undefined The Poisson distribution can be proven using the properties of the binomial distribution in the limit as the number of trials approaches infinity and the probability of success approaches zero, while the product of these two values remains constant. Specifically, as the number of trials (n) approaches infinity and the probability of success (p) approaches zero, with np being held constant, the binomial distribution converges to the Poisson distribution. This can be mathematically demonstrated through the limit definition and various probability theory concepts.