SIMULATION : THE POISSON DISTRIBUTION

The following simulation illustrates the concept of Poisson distribution, and its intimate connection with the binomial distribution.

A) Upper frame

    * The Poisson distribution corresponding to the displayed value of λ is the black graph.

    * The red graph is the B(n, p) binomial distribution for the displayed n and p values.

The relationship np =  λ is maintained throughout the animation (as long as the numbers make it is possible, see below).

    1) Change the value of n (λ remains constant). The value of the probability p changes so as to keep np = λ valid.

        Increase n and observe that the binomial distribution fits the Poisson distribution better for larger values of n (and therefore smaller values of p). This is in accordance with the very purpose of the Poisson distribution, which is to provide a good approximation of the binomial distribution for large n and small p.

    2) Change the value of p (while n is kept constant). Observe that the binomial distribution fits the Poisson distribution better for small values of p.

    3) Change the value of λ (while n is kept constant). the probability p is changed by the animation so as to preserve the np = λ relationship. If you increase λ, the probability p also increases, but of course, it cannot go beyond "1". Therefore, if you keep increasing λ beyond a certain value, no value of n can satisfy np = λ, and the binomial graph disappears. The Poisson graph is maintained, though.

When you stop increasing λ, a new binomial graph is displayed. The animation selects the smallest value of n that allows the np = λ relationship to be satisfied.

B) Lower frame

        As the purpose of the Poisson distribution is to approximate a binomial dsitribution, it is legitimate to ask how good is this approximation. The lower frame provides a graphical representation of the quality of this approximation.

    * Observe first that the binomial distribution is taller and narrower than the Poisson distribution with  λ = np (upper frame). Can you explain why ? If you hesitate, read the section on the variance of the Poisson distribution.

     * In the lower frame, the graph displays the ratio :

Ratio = P_Bin.{X = k} / P_Pois.{X = k}

k-values for which the Poisson distribution overestimates the binomial distribution are in black, whereas k-values for which the Poisson distribution underestimates the binomial distribution are in red.

    * The horizontal axis inside the green strip indicates the perfect match P_Bin.{X = k} et P_Pois.{X = k}.

    * On the right-hand side is the "Tolerance" control, that the user may activate step by step across a restricted range.

    * Now, a bar whose upper end is within the green strip is such that :

            (P_Bin.{X = k} / P_Pois.{X = k}) < Tolerance

or

            (P_Pois.{X = k} /P_Bin.{X = k}) < Tolerance

depending on the color of the bar.

They mark the k-values for which the Poisson approximation is within the preset tolerance level. Observe that the approximation is never at its best for the central part of the binomial distribution. The end-points of the binomial distribution are always poorly approximated.

So, within the range of values of the parameters accessible to the animation, the well approximated k-values are few, and the central zone of the binomial distribution is often poorly approximated.

This is because this animation is unfair to the Poisson distribution, which would normally be used with much lower values of p (10^-2 or less), and much larger values of n (100 or more). Then, the approximation is excellent in the central area of the binomial distribution, and over a large range on either side.

Nevertheless, keep in mind that the quality of the approximation degrades rapidly for k-values close to 0 or to n.