- #1

fluidistic

Gold Member

- 3,661

- 103

## Main Question or Discussion Point

I cannot seem to get the first moment of Poisson's distribution with parameter a: [itex]P(n_1)=\frac{a^{n_1}e^{-a}}{n_1!}[/itex] when using the characteristic function [itex]\phi _X (k)=\exp [a(e^{ik}-1)][/itex].

The definition of the first moment involving the characteristic function is [itex]<n_1>=\frac{i}{n} \frac{d \phi _X (k)}{dk} \big | _{k=0}[/itex].

I get [itex]<n_1>=\frac{1}{i} a(e^{ik}-1)aie^{ik}e^{a(e^{ik}-1)} \big | _{k=0}=0[/itex] because of the factor [itex]e^{ik}-1\big | _{k=0}=1-1=0[/itex].

However I should reach [itex]<n_1>=a[/itex].

I really do not see what I did wrong. I do know that the characteristic function is OK, according to wikipedia and mathworld. I also do know I should reach that the mean or first moment is worth "a" but I'm getting 0.

I've applied twice the chain rule and do not see any mistake, but obviously I'm doing at least one somewhere. Any help is welcome!

The definition of the first moment involving the characteristic function is [itex]<n_1>=\frac{i}{n} \frac{d \phi _X (k)}{dk} \big | _{k=0}[/itex].

I get [itex]<n_1>=\frac{1}{i} a(e^{ik}-1)aie^{ik}e^{a(e^{ik}-1)} \big | _{k=0}=0[/itex] because of the factor [itex]e^{ik}-1\big | _{k=0}=1-1=0[/itex].

However I should reach [itex]<n_1>=a[/itex].

I really do not see what I did wrong. I do know that the characteristic function is OK, according to wikipedia and mathworld. I also do know I should reach that the mean or first moment is worth "a" but I'm getting 0.

I've applied twice the chain rule and do not see any mistake, but obviously I'm doing at least one somewhere. Any help is welcome!