# Probability And Statistics | Week 6

## Probability And Statistics Week 6 Answers

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

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

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

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

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

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

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

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

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

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.

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