Week 6 : Assignment 6

Q1. Suppose that (X,Y) is bivariate discrete random variable such that the point (1,2) occurs with probability 1/4, (1,3) with probability 1/2, (2,3) and (3,1) with probabilities 1/8 each. Calculate P(Y=2|X=1).

a. 1/3

b. 1/4

c. 5/8

d. 3/4

Correct Answer : (A) 1/3

Q2. Let X and Y be two continuous random variables with the joint density function given by

f(,y)={ax+1, xy>0,x+y<1 0, otherwise.

Find P(Y < X).

a. 5/8

b. 1/8

c. 3/8

d. 1/4

Correct Answer : (A) 5/8

Q3. Suppose X and Y are two discrete random variables with the joint probability mass function given as in the following table:

Find Var (X).

(A) 0.8647

(B) 0.6784

(C) 0.4768

(D) 0.6875

Correct Answer : (D) 0.6875

Q4. Let X and Y be two continuous random variables with the joint density function

f(x,y)={6xy, 0<=x<=1,0<=y<=√x. 0, otherwise

Calculate E(Y|X = x)

(A) 2/3√x

(B) 1/3√x

(C) 1/2x

(D) x

Correct Answer : (A) 2/3√x

Q5. Consider following joint density functions f1(x1,y1) and f2(x2,y2) corresponding to the random variables (X1,Y1) and (X2,Y2) as

Which of the following statement is TRUE?

(A) (X1,Y1) is independent but (X2,Y2) is not independent

(B) (X1,Y1) is not independent but (X2,Y2) is independent

(C) Both (X1,Y1) and (X2,Y2) are independent

(D) Both (X1,Y1) and (X2,Y2) are not independent

Correct Answer : (A) (X1,Y1) is independent but (X2,Y2) is not independent 

Q6. Consider X and Y be two discrete random variables with joint mass function given as

Then find Cov(X,Y).

(A) 0.02595

(B) -0.02575

(C) 0.03165

(D) -0.03125

Correct Answer : (D) -0.03125

Q7. Suppose a fair die is rolled n times. Let X and Y be the random variables denoting the number of 1’s and number of 2’s respectively. Find n such that Cov(X, Y) = -1/4?

(A) n=4

(B) n =10

(C) n=9

(D) n=5

Correct Answer : (C) n=9

Q8. Suppose X and Y have bivariate normal distribution with parameters μx= 1, σx2 = 1, μy = 1, σy2 = 9, ρ = 1/2. Find P(X +Y > 0).

(A) 0.5123

(B) 0.7123

(C) 0.2877

(D) 0.4321

Correct Answer : (B) 0.7123

Q9. Suppose X and Y have bivariate normal distribution with parameters μx= 1,σx2 = 1, μy = -1, σy2 = 4, ρ = -1/2. Find a such that aX + Y and X + 2Y are independent.

(A) a=9

(B) a=8

(C) a=7

(D) a=6

Correct Answer : (C)  a=7

Q10.

Find E(X), E(Y), Var(X), Var(Y), ρ from the given probability density function.

(A) E(X) = 2, E(Y) = 1, Var(X) = 1, Var(Y) = 1, ρ = -0.8.

(B) E(X) = 2, E(Y) = 1, Var(X) = 1, Var(Y) = 1, ρ = 0.8.

(C) E(X) = 2,E(Y) = 1, Var(X) = 1, Var(Y) = 1, ρ = -1.6.

(D) E(X) = 2, E(Y) = 1, Var(X) = 1, Var(Y) = 2, ρ = 1.6.

Correct Answer : (B) E(X) = 2, E(Y) = 1, Var(X) = 1, Var(Y) = 1, ρ = 0.8.