Ononot — Let X, Y, Z be independent Poisson random variables with par

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Let $X, Y, Z$ be independent Poisson random variables with parameters $\lambda_1, \lambda_2, \lambda_3$ respectively. Define $U = X + Z$ and $V = Y + Z$ . What is the joint probability generating function $G_{U,V}(s, t) = E[s^U t^V]$ ?