# 18.9 Exercises

18.1 Suppose *X* and *Z* are independent Poisson random variables with means λ_{1} and λ_{2}, respectively. Let *Y* = *X* + *Z*. Demonstrate that *X*|*Y* = *y* is binomial.

18.2 Suppose *X* is binomially distributed with parameters *n*_{1} and *p*, that is,

Suppose also that *Z* is binomially distributed with parameters *n*_{2} and *p* independently of *X*. Then *Y* = *X* + *Z* is binomially distributed with parameters *n*_{1} + *n*_{2} and *p*. Demonstrate that *X*|*Y* = *y* has the hypergeometric distribution.

18.3 Consider a compound Poisson distribution with Poisson mean λ, where *X* = *Y*_{1} + ··· + *Y*_{N} with E(*Y*_{i}) = μ_{Y} and Var(*Y*_{i}) = σ^{2}_{Y}. Determine the mean and variance of *X*.

18.4 Let *X* and *Y* have joint probability distribution as given in Table 18.2.

**(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(

*X*

^{2}|

*y*). and Var(

*X*|

*y*) for

*y*= 0, 1, 2.

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

**(b)**The marginal pdf is

**(c)**The variables ...

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