# Probability And Statistics | Week 6

## Probability And Statistics Week 6 Answers

**Link : Probability And Statistics (nptel.ac.in)**

**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 byf(,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 functionf(x,y)={ 6xy, 0<=x<=1,0<=y<=√x.{ 0, otherwiseCalculate 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 μ _{x}= 1, σ^{x2} = 1, μ_{y} = 1, σ^{y2} = 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 μ _{x}= 1,σ^{x2} = 1, μ_{y} = -1, σ^{y2} = 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.