Due 12/14/98

Complete the following problems.

1.

Suppose that the random variable (R,S) is uniformly distributed on the set $\{x^2 + y^2 \leq 1\}$. Find

(a)

The distribution function of $\sqrt{R^2 + S^2}$;

(b)

The distribution function of R.

(c)

The conditional density of S given R = r.

(d)

E[S | R = r].

2.

Suppose that the random variable (R,S) is such that R is uniformly distributed on (0,1) and that the conditional distribution of S given R = r is binomial (10,r). (This would be the situation for an experiment where a number R was chosen at random in (0,1) and a coin with probability R of coming up heads was tossed 10 times, with S being the resulting number of heads.) Find the probability mass function, mean and variance of S.

3.

Suppose that the random variable R is uniformly distributed on (0,1). Find the distribution function of $-\log(R)$.

4.

Suppose that E[R] = 1, ${\rm Var}(R) = 2$ and E[R³] = 3. Find E[(R-2)³].

5.

A chest has three drawers. One drawer contains two gold coins, one drawer contains a gold coin and a loaded mouse trap, and the third drawer contains two loaded mouse traps. The lights are turned off, you are blindfolded, a drawer is chosen at random and a gold coin is removed. What is the probability that when you reach into the drawer you will retrieve another gold coin?

6.

Suppose that V is a random sample of size k from a Gamma distribution with $\alpha = 2$ and $\beta$ unknown. Compute the maximum likelihood estimate of $\beta$. Is this an unbiased estimate of $\beta$?

7.

Suppose that in the city of Milwaukee there are 50,000 people who intended to vote for John Norquist for mayor, and 49,000 people who intended to vote for Richard Artison. Before the election, a group of 101 people was chosen and asked which of the two candidates they would vote for. What is the probability that the majority of this group would support Richard Artison? You should assume that all groups of 100 are equally likely to be chosen.

8.

In the previous question, suppose that 101 people out of the 99,000 were asked who they would vote for, and no effort was made to prevent the same person from being asked more than once. What is the probability that the majority of the 101 responses would favor Artison?