Loss Models: From Data to Decisions, 4th Edition

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,

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 + ··· + Y_N with E(Y_i) = μ_Y and Var(Y_i) = σ²_Y. 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

Show the following:

(a) The conditional density function is

Hence,

(b) The marginal pdf is

(c) The variables ...