What type of distribution for epsilon? can I generate observations for epsilon ?

Question

$X$ has zero truncated poission distribution ($\lambda$)

then pmf is $f(x)=\frac{\lambda^x}{(e^\lambda-1)x!}$

and the probability generating function is $\phi_X(s)=\frac{e^{\lambda s}-1}{e^\lambda-1}$,

Now i have anthor variable $\epsilon$ has the probability generating function

$\phi_\epsilon(s)=\frac{\phi_X(s)}{\phi_X(1-\alpha+\alpha s)}=\frac{e^{\lambda s}-1}{(1-e^\lambda)e^{\lambda(1-\alpha+\alpha s)}}$

and then $\epsilon$ has the pmf as $P(\epsilon=j)=\frac{[(1-\alpha)^j-(-\alpha)^j]\lambda^j e^{\lambda \alpha}}{j! (e^\lambda-1)}$......  λ>0 and 0≤α≤1/2

What type of distribution for epsilon? can I generate observations for epsilon ?

Answer 1

This is a partial answer on the part of generating observations for ε. I am not sure about what kind of distribution epsilon is.

Since your ε has only non-negative integer values, you can derive the cumulative distribution function from probability generating function,

$\phi_\epsilon(s)/(1-s)$. It is not hard to see. You expand $1/(1-s)=1+s+s^2+\dots$ and collect the coefficients of $s^n$ in $\phi_\epsilon(s)/(1-s)$. The coefficient is the probablity $P(ε\leq n)$.

Once you have CDF, generating observations is standard by using unitary distribution on [0,1] and the inverse function of CDF.