# Simulation of a random variable given the moment generating function after exponential tilt

The random variable $S$ follows a distribution with moment generating function $$M_S(v)=\frac{\beta\mu v}{1+(1+\beta)\mu v-M_X(v)}$$ I have been looking in some books about this m.g.f and I found that the distribution of this random variable is called compound geometric, but I have not found detailed information about the Probability Distribution of this.

I need to simulate a random variable $S_{\theta}$ whose moment generating function is: $$M_{S_\theta}(v)=\frac{M_S(v+\theta)}{M_S(\theta)}$$

Since I don't know exactly the distribution of $S$ I can't generate the random variable $S_\theta$. I only know that $P(S\leq u)=1-\Psi(u)$ that is a value that I can compute becase I have an expresion for $\Psi$.

How to proceed?

• What's the distribution of the components? Why do you want to start from the mgf? – Glen_b Apr 20 '18 at 11:12
• But I cannot see how to generate the random variable with such moment generating function, I am applying the exponential tilting technique. – Boris Apr 20 '18 at 14:38
• Again, to return to my initial comment, it's not clear why you would try to simulate using the mgf -- it's easy to simulate directly. Why is it necessary to use the mgf to simulate? (That's not to suggest that such a task is not possible, at least approximately, but it seems to be completely unnecessary in this case) – Glen_b Apr 21 '18 at 1:32
• I can generate S but how to generate $S_\theta$? – Boris Apr 23 '18 at 22:30
• There appears to be a closely related question here – Glen_b Apr 24 '18 at 2:36