4. (a) For the hierarchical model Y|A Poisson(A) and A Gamma(a,b), find the marginal distribution, mean, and variance of Y. Show that the marginal distribution of Y is a negative binomial if a is an integer. (b) Show that the three-stage model YA ~ Binomial(N,P), NA ~ Poisson(A), and A ~ Gammala, B) leads to the same marginal distribution of Y.
2. Let X and Y are independent Poisson random variables with parameter and pe respectively. (a) Show that X + Y is a Poisson random variable with parameter +H. (b) What is the conditional probability P(X = X(X + Y = n). Show your steps. (c) Hence, are X and X+Y independent?
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You are given a fair die, i.e. the outcomes of any throw are contained in the sample space 
(a) Derive the joint pdf of X and Y. Provide this in the form of a table
Y=0 Y=1
| X=0 | ||
| X=1 |
b) What is the marginal probability distribution of Y?
(c) What is the conditional probability distribution of X, given that Y = 1?
(d) What is E (X|Y = 1)?
(e) What is Cov (X, Y )?
(f) Are X and Y independent random variables?
A salesperson makes four calls per year. A sample of 100 days given the following frequencies of sales volumes.
| Number of Sales | Observed Frequency (days) |
| 0 | 30 |
| 1 | 32 |
| 2 | 25 |
| 3 | 10 |
| 4 | 3 |
| TOTAL | 100 |
Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial distribution. The Binomial distribution is represented by:
For this exercise, assume that the population has a binomial probability distribution with n=4, p=0.30 and x= 0, 1, 2, 3 and 4.
(a) Compute the expected frequencies for x=0, 1, 2, 3 and 4 by using the binomial probability function. Combine categories if necessary to satisfy the requirement that the expected frequency is five or more for all categories.
(b) Use the goodness of fit test to determine whether the assumption of a binomial probability distribution should be rejected. Because no parameters of the Binomial probability distribution were estimated from the sample data, the degrees of freedom are k-1 .
