18.1 Suppose X and Z are independent Poisson random variables with means λ1 and λ2, respectively. Let Y = X + Z. Demonstrate that X|Y = y is binomial.
18.2 Suppose X is binomially distributed with parameters n1 and p, that is,
Suppose also that Z is binomially distributed with parameters n2 and p independently of X. Then Y = X + Z is binomially distributed with parameters n1 + n2 and p. Demonstrate that X|Y = y has the hypergeometric distribution.
18.3 Consider a compound Poisson distribution with Poisson mean λ, where X = Y1 + ··· + YN with E(Yi) = μY and Var(Yi) = σ2Y. Determine the mean and variance of X.
18.4 Let X and Y have joint probability distribution as given in Table 18.2.
18.5 Suppose that X and Y are two random variables with bivariate normal joint density function
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