Given the moment generating function m(t) = e3(e^t-1), find P(X=0), the first and second moments of X, and Var(X).

The given moment generating function (MGF) is $m(t) = e^{3(e^t-1)}$.
This is the MGF of a Poisson distribution with parameter $\lambda$. The standard MGF for a Poisson distribution is $M_X(t) = e^{\lambda(e^t-1)}$.
By comparing the given MGF with the standard form, we can identify that $\lambda = 3$.
Thus, $X$ follows a Poisson distribution with parameter $\lambda = 3$.
For a Poisson distribution $X \sim \text{Pois}(\lambda)$:

• The probability mass function is $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
• The first moment (mean) is $E[X] = \lambda$
• The second moment is $E[X^2] = \lambda + \lambda^2$
• The variance is $Var(X) = \lambda$