Question 1. The density of (X, Y ) is given by

f (X, Y) = C(X² + Y), 0 < Y < 1, -√Y< x < √Y.

(a) What is the constant C?

(b) What is the density of Y ?

(c) What is the density of X given Y = Y?

(d) What is P (X² ≥ 1/4|Y ≤ 3/4)?

(e) Are X and Y independent (and why)? (f) What is P (X > Y³)?

Question 2. In one year's probability class, 10% of students received a final mark between 95 and 100. Assume 100 student marks are chosen with replacement from the subject and let X be the number in the sample that are between 95 and 100.

(a) What is the expected value of X?

(b) What is the variance of X?

(c) Compute exactly the probability X is between 7 and 13 (inclusive).

(d) Approximate the previous probability using both a Poisson and a normal distribution.

(e) Comment on which approximation in part (d) is better and whY.

Question 3. Let U have density (5/2)u⁴ on the interval (-1, 1).

(a) Find the distribution function and density of T = U⁵.

(b) Find the distribution function and density of V = 1 - U².

(c) What is the mean and variance of V = 1 - U²?

Question 4. Roll a die repeatedly and let X be the number of rolls before the first time a five or six appears (so it is possible for X to be zero). Now toss a fair coin X times and let Y be the number of heads.

(a) What is the joint probability mass function of (X, Y )?

(b) Find the probability mass function of Y and name the distribution of Y .

(c) Find the probability mass function of X|Y = 3 and name this distribution.

Question 5. Let (X, Y) be uniformly chosen on the square region {(X, Y) : |X| + |Y| < 1}.

(a) Write down the density of (X, Y ).

(b) What is the distribution function of the angle 0 < Θ < 2Π made between the positive X-aXis and the raY connecting the origin to the point (X, Y )?

(c) What is the density of Y ?

(d) What is the density of X|Y = y for -1 < y < 1?