Question: Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be independent Poisson random variables with means \(\lambda_{1}, \lambda_{2}, \ldots\) respectively. Find the

1. probability function of \(\sum_{i=1}^{n} Y_{i}\).
2. conditional probability function of \(Y_{1}\), given that \(\sum_{i=1}^{n} Y_{i}=m\).
3. conditional probability function of \(Y_{1}+Y_{2}\), given that \(\sum_{i=1}^{n} Y_{i}=m\).

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