If the probability-generating function $G_{S_{i}}(u+v)$ satisfies the functional equation $$\begin{align*} G_{S_{i}}(u+v) & =G_{S_{i}}(u) G_{S_{i}}(v) \end{align*}$$ and exhibits the characteristics of an exponential function, where a function $f(x)$ satisfying $f(x+y)=f(x)f(y)$ can be represented as $f(x)=e^{kx}$, can I infer that the random variable $S_i$ follows a Poisson distribution? If so, could you please provide a proof for this claim?
2026-04-04 02:12:45.1775268765
What does it mean for a probability generating function to be an exponential function
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