Question: Let \(X\) have Poisson \((\lambda)\) distribution and let \(Y\) have Poisson \((2 \lambda)\) distribution.

1. Prove \(P(X \geq Y) \leq \exp (-(3-\sqrt{8}) \lambda)\) if \(X\) and \(Y\) are independent.
2. Find constants \(A<\infty, c>0\), not depending on \(\lambda\), such that, without assuming independence, \(P(X \geq Y) \leq A \exp (-c \lambda)\)

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