18.9 Exercises

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,

equation

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.

Table 18.2 Data for Exercise 18.4.

(a) Compute the marginal distributions of X and Y.
(b) Compute the conditional distribution of X given Y = y for y = 0, 1, 2.
(c) Compute E(X|y), E(X2|y). and Var(X|y) for y = 0, 1, 2.
(d) Compute E(X) and Var(X) using (18.3), (18.6), and (c).

18.5 Suppose that X and Y are two random variables with bivariate normal joint density function

equation

Show the following:

(a) The conditional density function is
Hence,
(b) The marginal pdf is
(c) The variables ...

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