Questions -

Q1. Provide an answer to each of the following questions.

(i) Let X and Y be continuous random variables with continuously differentiable CDF F_X,Y  on all of R². Define the joint PDF as

f_X,Y(X,Y) = ∂²F_X,Y(x, y)/∂x∂y,    (x, y) ∈ R².

Show that if X and Y are independent, that is, F_X,Y(x, y) = F_X(x)F_Y(y) for all x and y, then f_X,Y(x,y) = f_X(x)f_Y(y).

(ii) Let X₁, X₂, and X₃ be independent Normal (1, 2) random variables. Find E[X₁ + 3X₂² exp(-X₃)].

(iii) Let X ∼ Poisson(2) and let Y|X = x ∼ Normal(3x,1 + x²), x ∈ {0, 1, 2, ... }. Find E(Y) and Var(Y).

(iv) For continuous random variables X₁ and X₂, assume the joint MGF, ψ(t₁, t₂), is finite in a neighborhood of 0. Show that

E(X₁X₂²) = ∂ψ³(0, 0)/∂t₁∂t₂².

Q2. Let (X, Y) be a bi-variate random vector with support (0, 1) x (0, 1). Its joint PDF is

f_X,Y(x, y) = 1 + a(1 - 2x)(1 - 2y), (x, y) ∈ (0,1) x (0,1) = 0 otherwise

where |α| ≤ 1.

(i) Verify that f_X,Y is a valid PDF.

(ii) Find the CDF associated with f. (Hint: Be careful here. You have to handle cases where (x, y) is not in the unit square.)

(iii) Show that X has a marginal Uniform(0, 1) distribution. What about Y?

(iv) Find P(X ≤ Y) when α = 1.

(v) For what value(s) of α are X and Y independent?

(vi) Find Corr(X, Y). [Hint: You might want to compute E(XY).]

(vii) What is the maximum possible value for the correlation across all values of a?

Q3. (i) Show that the moment generating function for the Binomial(n, p) distribution is

ψ(t) = [1 - p +p exp(t)]ⁿ.

(Hint: It is helpful to use the fact that every binomial distribution sums to unity across its support.)

(ii) Let y_j ∼ Binomial(n_j, p) for j = 1,. . . . ,m. Define

W = Y₁ + Y₂ + · · · + Y_m

What is E(W)?

(iii) If the {Y_j} are pairwise uncorrelated, what is Var(W)?

(iv) If the {Y_j} are independent, what is the distribution of W?

(iv) Now let {X_i : i = 1, . . . , n} be independent Bernoulli(p) random variables. Find the (discrete) joint PDF of (X₁, X₂, · · · ,X_n).

(v) In the setting of part (iv), let Y = X₁ + · · · + X_n. Show that the joint PDF of (X₁, . . . ,X_n) given Y = y is

Q4. Let X_j, j = 1, . . . , k denote Bernoulli(p_j) random variables such that

X₁ + X₂ + · · · + X_k = 1.

(i) Explain why this setup can describe situations where one can choose a single option from k exhaustive and mutually exclusive options. How come the Xi cannot be independent?

(ii) Show that it must be the case that p₁ + p₂ + · · · + p_k = 1.

(iii) Derive the k x k variance-covariance matrix of the vector X = (X₁, X₂, . . . ,X_k)'. [[Hint: First argue that X_hX_j = 0 when h ≠ j.]

Q5 (i) Let X be an m x 1 random vector with finite second moments, and assume Corr(X) has an equi-correlation structure with common correlation ρ. Use the fact that the correlation matrix R is PSD and study the quadratic form j'_mRj_m with j'_m = ( 1 1 · · · 1 ) to conclude ρ ≥ -(m-1)^-1.

(ii) Let Z be an m x 1 random vector with E(Z) = 0 and Var(Z) = I_m and define

X = Z - j_mZ^- = QZ,

where Z^- = m^-1 _j=1∑^m Z_j and

Q = I_m - j_m(j'_mj_m)^-1j'_m

is the matrix that demeans an m x 1 vector. Show that Corr(X) has an equicorrelation structure with ρ = - (m - 1)^-1.