List of top Statistics Questions on Poisson distribution

Let \( X \) and \( Y \) be discrete random variables with joint probability mass function \[ p_{X, Y}(m, n) = \frac{\lambda^n e^{-\lambda} 2^n m! (n - m)!}{n!}, \quad m = 0, \dots, n, \quad n = 0, 1, 2, \dots, \] where \( \lambda \) is a fixed positive real number. Then which one of the following options is correct?

Let \( \{ N(t): t \geq 0 \} \) be a homogeneous Poisson process with the intensity/rate \( \lambda = 2 \). Let} \[ X = N(6) - N(1), \quad Y = N(5) - N(3), \quad W = N(6) - N(5), \quad Z = N(3) - N(1). \] Then which one of the following options is correct?
Let \( X \) be a random variable having the Poisson distribution with mean \( \log_e 2 \). Then \( E\left( e^{(\log_e 3)X} \right) \) equals:
Let \( X \) be a random variable having Poisson distribution with mean \( \lambda>0 \). Then \( E \left( \frac{1}{X+1} \mid X>0 \right) \) equals: