STAT 450

Problems: Assignment 2

Basic Questions

1. Suppose X is Poisson(). After observing X a coin landing Heads with probability p is tossed X times. Let Y be the number of Heads and Z be the number of Tails. Find the joint and marginal distributions of Y and Z.

2. Suppose X and Y are independent with and . Let Z=X+Y. Find the distribution of Z given X and that of X given Z.

3. Question 31 on page 216.

4. Question 33 on page 216.

5.   Question 46 on page 216.

6.   Question 55 on page 217.

   Bonus Questions: if you do these you need do only questions 5 and 6 above for full marks on the assignment.

7. Suppose X and Y have joint density . Prove from the definition of density that the density of X is .

8. Let be the bivariate normal density with mean 0, unit variances and correlation and let be the standard bivariate normal density. Let .
   1. Show that p has normal margins but is not bivariate normal.

   2. Generalize the construction to show that there rv's X and Y such that X and Y are each standard normal, X and Y are uncorrelated but X and Y are not independent.

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