Suppose the moment generating function of the random variable $X$ is $1-p + pe^{tN}$ for some parameter $N$ (which can be assumed to be large) and $p \in (0,1)$. How would I go about calculating the distribution of $X$?
Would it be correct to say $X$ is approximately a $Binomial(N,p^{1/N})$ distribution, which has mgf: $(1-p^{1/N} + p^{1/N}e^{t})^N$? What would the approximation error be?
(I actually need to calculate the expectation of $1/X$ and other related quantities, but knowing the full probability distribution approximately would be useful).
A Bernoulli random varianble (r.v.) is $\Pr(X=0)=1-p$ and $\Pr(X=1)=p$. Take $Y=X\times N$ where here $N$ is a specified scalar. Then $\Pr(Y=0)=1-p$ and $\Pr(Y=N)=p$. The Moment Generating Function (MGF) will be
$\sum_{Y=y}e^{tY}\Pr(Y=y)= (1-p)e^{t0} + pe^{tN}.$
This is the MGF specified by the OP.