Problem 1:

Suppose Y is a rv with pdf f(y) = ky, y = 3/n, 6/n, 9/n, ..., 3n/n.

(a) Find the value of the constant k and write down Y's cdf.

(b) Find simple general expressions for EY, VarY, P(Y = 3/2) and P(Y > 3/2).

(c) For the case n = 10, evaluate EY, VarY, P(Y = 3/2) and P(Y > 3/2), and write down and sketch Y's pdf and cdf.

(d) Repeat (c) for the limiting case as n tends to infinity. [5 marks]

Hint: The rv in (d) has a cdf equal to the limit of the cdf in (a) as n → ∞.

Problem 2:

Suppose the rv Y is normally distributed with mean -2 and variance 25.

(a) Find the probability that Y exceeds 1, given that Y is positive.

(b) Find the expected value of Y, given that Y is positive.

Hint: For (b), first derive the cdf of the rv given by X = (Y | Y > 0).

Problem 3:

(a) Suppose that a rv Y has mgf m(t) = e^{k(et -1)/}(1 - bt)^a

Differentiate this mgf twice and thereby obtain the mean and variance of Y.

(b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t),

i.e. r(t) = logm(t). Find r'(t) and r''(t) , and express r'(0) and r''(0) in terms of EW and VarW.

(c) Use the result in (b) to find the mean and variance of the rv Y in (a).

(d) Find the mean and variance of the rv Y in (a) using a method which does not involve any differentiation.

Problem 4:

You wish to fence off a rectangular paddock on one side of a river running through your property in a straight line. No fence is required along the side of the paddock formed by the river. The fence you will use is rolled up in a shed, and you are at the moment not quite sure how long it is. However, you are certain that it is between 3 and 5 km long, and your uncertainty regarding its length can be represented by a parabolic probability density function which tapers off to zero at 3 and 5 kms.

(a) Find and sketch the probability density function of Y, the area of the largest paddock you will be able to fence off.

(b) Find the expected value, mode and median of Y. Then illustrate these three quantities in the figure in (a).

Problem 5:

Consider two rvs X and Y with joint pdf f (x, y) = k - y, 0 < y < x² <1.

(a) Sketch the region in two dimensions where f(x,y) is positive. Then find the constant k and sketch f(x,y) in three dimensions.

(b) Find and sketch the marginal pdf f(x), the conditional pdf f(x|1/2), and the conditional cdf F(y|1/2).

(c) Find P( X < Y | Y >1/ 2), E( X | Y =1/ 2) and E( X | Y >1/ 2).

(d) What is the correlation between X and Y? Are X and Y independent? Justify your answer.

(e) Find the value of the pdf of U = X + Y evaluated at u = 0.8. Hence, or otherwise, estimate P(0.8 < X +Y < 0.801).