Workshop: Poisson Distribution

STUDENT NAME:

Problem #1:  Consider the Poisson Distribution of the form

μ = 0.9;
P[n_] :=  (E^(-μ)) ((μ^n)/(n!))

Graph your result over the range {n, 0, 10} using Plot. You might need to use the option  PlotRange->All to make sure you see the whole range.

Your Plot above is a continuous function of n which is a little strange since n is considered to be an integer and also it might be unclear to you how to calculate n! if n is NOT an integer.  To avoid these problems, you can graph only the n=integer value using (1) the Table function and (2) using ListPlot.

data = Table[{n, P[n]}, {n, 0, 10}];
P1 = 
      ListPlot[data, PlotStyle -> PointSize[0.02], 
        AxesLabel -> {"n", "P[n]"}]

The label  P1 is assigned to this graph for later use in problem #4.

Problem #2:  Calculate the sum

(∑_(n = 0)^10) P[n]

What value did you obain?  Can you explain this value theoretically?

Problem #3:  Calculate the average number of steps  \(TraditionalForm\`n\&_\)  to the right using

\(n\&_ = \ ∑\+\(n = 0\)\%10 n*P[n]\)

What value did you obain?  Explain the value \(TraditionalForm\`n\&_\) theoretically.

Problem #4:  Consider the differrent Poisson Distribution with μ having a different value

μ = 2.5;
P[n_] :=  (E^(-μ)) ((μ^n)/(n!))

Graph your result over the range {n, 0, 10} using Plot. Assign the label P2 to this graph for later use.  Make the plot points a little larger than in problem #1 in order to distinguish them.  Use Show[P1,P2] to graph this Plot and the one of problem #1 together.  When μ is larger, qualitatively how does the shape of the graph change?

Calculate the average number of steps to the right \(TraditionalForm\`n\&_\) using

μ =  (∑_(n = 0)^10) n*P[n]

What value did you obain?  Explain the value you got for \(TraditionalForm\`n\&_\) ?

Problem #5:   Calculate the average of the square of the deviations of the number of the steps to the right (σ^2) using

σ2 = (∑_(n = 0)^10)(((n -  μ ))^2)*P[n]

What numerical value did you obtain for (σ^2)?  The theoretical value for (σ^2) is

(σ^2)= μ

Does your numerical value for (σ^2) agree with the theoretical prediction?