Questions -

Q1. Let X be the number of heads appearing when a fair coin is flipped give times. Consider predicting X using three different values: 2, 5/2, and 3.

(i) If you want to minimize the mean squared error, which of the three values would you use? Find the MSE for each of the three predictions.

(ii) If you want to minimize the mean absolute error, which value should you use? Find the MAE for each of the three predictions.

(iii) If your prediction rule is to use a mode to predict X, what would you predict?

Q2. The PDF for the Pareto(α, β) distribution is given by

f(x) = α/β(1+x/β)^α+1, x > 0           

f(x) = 0, x ≤ 0

(i) Find the CDF and the survival function.

(ii) Use the survival function to show that if α > 1 then mean is finite and E(X) = β/(α - 1). What happens if α ≤ 1?

(iii) Find the median of X.

(iv) Is E(X) > Med(X) always for this distribution? Justify your answer.

(v) Find Med[log(X)].

Q3. Provide an answer to each of the following questions.

(i) Let X be a continuous random variable with support on (0, ∞), with CDF F_x differentiable on (0, ∞). Let f_x be the PDF of X. Define Y = 1/√x. Find the PDF of Y, f_y, in terms of f_x.

(ii) Let X be a continuous random variable with support (a, b) for -∞ < a < b < ∞. Show that a ≤ E(X) ≤ b.

(iii) Suppose that the MGF of X, ψ_x, is finite on R. For real numbers a and b, let Y = a + bX. Show that the MGF of Y is ψ_y(t) = exp(at)ψ_x(bt).

(iv) Let X ∼ Uniform(0, 1). Is E(X^-1) < ∞?

Q4. (i) Let X ∼ Normal(μ, σ²). What is the distribution of -X?

(ii) What are the distributions of exp(X) and exp(-X)?

(iii) The hyperbolic sine and cosine functions are defined on R by

sinh(x) = [exp(x) - exp(-x)]/2

cosh(x) = [exp(x) + exp(-x)]/2

For X ∼ Normal(μ, σ²), define Y = sinh(X). Show that E(Y) = exp(σ²/2) sinh(μ).

(iv) Find E(Y²) in terms of cosh(2μ).

Q5. (i) Let X ∼ Geometric(p) for 0 < p < 1 where the support of X is {1, 2, ... }. Find P(X ≤ k) for any positive integer k.

(ii) For X in part (i), find Var(X) as a function of p. (You should provide a derivation of some sort.)

(iii) Consider the St. Petersberg paradox, where a fair coin is flipped until it comes up heads, and the payout is W = 2^k if k is the first trial where a head occurs. If you pay $100 to play the game, what is the probability that you lose money?

(iv) In the setup from part (iii), suppose your utility function u(w) = √w. Find the expected utility from the payout (not net of the $100 fee).