Moment-generating Function of Poisson Distribution

Date: 10/25/1999 at 10:13:27
From: Goksel Ozdilek
Subject: Moment-generating Function of Poisson Distribution

Dear Dr. Math,

I have two questions:

1. The moment-generating function of a Poisson distribution is given 
   by

     M.G.F. (s,t) = e^(lambda(s-1)t)

   What does this moment generating function imply? (Is lambda*t the 
   intensity?)


2. mij = 1 + Sk 1 j  Pik mkj

   {mu ij = 1 + SIGMA (k is not equal to j) Pik * mu kj}

   How can I derive the expectation from this formula?

   E [Tij] = E [Tj | Xo = i]

   Tij first passage time from i to j

Thanks,
Goksel

Date: 10/25/1999 at 14:31:20
From: Doctor Anthony
Subject: Re: Moment Generating Function of Poisson Distribution

1. The moment generating function is 

     M(t) = Expected value of e^(xt)

          = SUM[e^(xt)f(x)]

and for the Poisson distribution with mean a

            inf
          = SUM[e^(xt).a^x.e^(-a)/x!]
            x=0

                   inf
          = e^(-a).SUM[(ae^t)^x/x!]
                   x=0    

          = e^(-a).e^(ae^t)

          = e^[a(e^t -1)]

The mean of the distribution is the coefficient of t/1! and E(x^2) is 
the coefficient of t^2/2! in expansion of the MGF as a series.

     M(t) = 1 + a(e^t -1) + a^2(e^t -1)^2/2! + ...

          = 1 + a(t + t^2/2! + ...) + a^2(t + t^2/2! + ...)^2/2! + ...

          = 1 + a(t + t^2/2! + ...) + a^2(t^2 + terms higher than
                                                         t^2)/2! + ...

From this we see that coefficient of t/1! = a   (so mean = a)

Coefficient of t^2/2! = a + a^2 

Therefore

     E(x^2) = a + a^2

and 

     var(x) = E(x^2) - mean^2

            = a + a^2 - a^2

            =  a

Therefore mean and variance are both equal to a.


2. I'm afraid I cannot follow your notation here. However the method 
is to expand the MGF as a series and then find the coefficient of t. 
This will give you the mean.  The coefficient of t^2/2! will give you 
E(x^2) and then variance = E(x^2) - mean^2.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/