2 Marks Questions and Answers

UNIT II

TWO DIMENSIONAL RANDOM VARIABLES

'2 Marks' Questions and Answers

 

1. Show that Cov² (X, Y) ≤ Var (X). Var (Y)  [A.U. N/D 2004]

Solution: Let X and & Y be 2 R.V's. For any real number ɑ,

This is a quadratic in 'a' and is always non-negative so the discreminent must be non-positive.

 [Cov (X, Y)]² ≤ Var(X). Var (Y)

 

2. Find the acute angle between the two lines of regression.

[A.U A/M 2019 (R8) RP] [A.U. AM 1003) [A.U. A/M 2011]

Sol. Angle between the lines, is given by

 

3. If X and Y are independent random variables with variance 2 and 3, then find the variance of 3X + 4Y.

 [A.U. A/M 2003, M/J 2013] [A.U A/M 2018 R-08]

Sol. R.V's X and Y are independent R.V's with variance 2 and 3.

(i.e.,) Var (Y) = 2 and Var (Y) = 3

Consider Var (3X + 4Y)

Var (ax) = a² Var (X).

= 3² Var (X) + 4² Var(Y) = 9 × 2 + 16 × 3 = 18 + 48 = 66

 

4. If two random variables X and Y have probability density function (p.d.f) f(x, y) = ke^-(2x+y) for x, y > 0. Find 'k'.

Solution :

By the property of the joint pdf,

 

5. If the joint pdf of (X, Y) is f(x, y) =  find P (x + y ≤ 1)  [A.U. N/D 2005] [A.U M/J 2016 R13 (RP))

Solution :

 

6. Determine the value of the constant c if the joint density function of two discrete random variables X and Y is given by p (m, n) = c mn, m = 1, 2, 3 and n = 1,2,3  [A.U N/D 2015, R-8] [A.U A/M 2017 R-8]

Solution:

Given: p (m, n) = c m n, m = 1, 2, 3 and n = 1, 2, 3

 

7. What do you mean by correlation between two random variables? [A.U A/M 2015 K8]

Solution

Degree of relationship and nature of relationship.

 

8. The joint probability mass function of a two dimensional random variable (X, Y) is given by p (x, y) = k (2x + y), x = 1, 2 and y = 1, 2, where k is a constant. Find the value of k. [A.U N/D 2015 R-13]

Solution:

Given: P (x,y) = k (2x + y)

x = 1, 2 and y = 1,2

 

9. Let (X, Y) be a two-dimensional random variable. Define co-variance of (X, Y). If X and Y are independent. What will be the covariance of [ (X, Y)? [A.U M/J 2016, R-13 RP]

Solution:

Cov. (X, Y) = E [(X-E (X)) (Y - E (Y))]

= E [XY]-E[X] E [Y]

If X and Y are independent, then E [XY] = E[X]E[Y]

Cov. (X, Y) = E[X]E[Y] - E [X]E[Y] = 0

 

10. Can y = 5 + 2.8x and x = 30.5y be the estimated regression equation of y on x respectively explain your answer.

Solution :

b_yx = 2.8, b_xy = -0.5

r = ± √(b_xy) (b_yx) = imaginary

They can not be estimated regression equations.

 

11. The joint probability density function of the random variable x and  is defined as

Find the marginal PDF's of x and y.

Sol.

 

12. Let X and Y be two independent R.Vs with Var(X) = 9 and Var(Y) = 3.

Find Var(4X - 2Y + 6).

[A.U M/J 2016 R13 (PQT)] [A.U N/D 2019 (R17) PQT)

Solution: Given: Var (X) = 9, Var (Y) = 3

Var [4X - 2Y + 6] = 4² Var (X) + (-2) ² Var (Y)

= (16) (9) + (4) (3)

= 144 + 12 = 156

 

13. The joint probability density function of (X, Y) is

Calculate P (X ≤ 2Y)

 [A.U A/M 2019 (R17) PQT]

Solution :

 

14. Define covariance and coefficient of correlation between two random variables x and y.

[A.U A/M 2019 (R17) RP]

Solution:

Cov. (X, Y) = EXY - EX]E[Y]

r (X, Y) = Cov (X, Y) / σ_X = σ_Y

 

15. The joint pdf of a bivariate random variable (X, Y) is given by

 where k is a constant. Determine the value of k.

 [A.U A/M 2019 (R17) RP]

Solution:

 

16. Prove that the correlation coefficient P_XY of the R.V's X and Y takes value in the range -1 and 1.

Solution: