Important 2 marks Questions and Answers

Important '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 a,

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

2. The lines of regression in a bivariate distribution are x + 9y = 7 and y + 4x = 49/3. Find the co-efficient of correlation. [A.UD 2015-RS] [A.U A/M 2017 R-08]

Solution:

The co-efficient of correlation between x and y is

3. The two equations of the variables X and Y are x = 19.13 - 0.87 y and y = 11.64 - 0.50 x. Find the correlation co-efficient between X and Y. [AU, May, '99]

Solution:

The co-efficient of correlation between x and y is

4. Find the acute angle between the two lines of regression. [A.U A/M 2019 (R8) RP] [A.U. A/M 2003]

Solution :

Angle between the lines, is given by

5. 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]

Solution:

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

6. State the equations of the two regression lines. What is the angle between them? [A.U. N/D 2003] [A.U Trichy A/M 2010] [A.U CBT M/J 2010] [A.U N/D 2017 R-08]

Solution 

7. State the basic properties of joint distribution of (X, Y) when X and Y are random variables.

Solution:

8. If two random variables X and Y have probability density function (p.d.f) f(x, y) = k e^-(2x+y) for x, y > 0. Find 'k'. [A.U. A/M 2005] [A.U N/D 2016 R13 (PQT)][A.U. N/D 2005]

Solution :

By the property of the joint pdf,

9. 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 :

10. 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

11. What do you mean by correlation between two random variables? [A.U A/M 2015 R8]

Solution

Degree of relationship and nature of relationship.

12. 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)

13. 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:

14. Can y = 5 + 2.8x and x = 3 - 0.5y be the estimated regression equation of y on x respectively explain your answer.

Solution :

.'. They can not be estimated regression equations.

15. The joint probability density function of the random variable x and y is defined as f(x,y) =  Find the marginal PDF's of x and y [A.U N/D 2016 R-13, RP]

Solution :

16. 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

17. The joint probability density function of (X, Y) is  Calculate P(X ≤ 2Y) [A.U A/M 2019 (R17) PQT]

Solution

18. Define covariance and coefficient of correlation between two random variables x and y. [A.U A/M 2019 (R17) RP]

Solution:

19. 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:

20. Prove that the correlation coefficient ?xy of the R.V's X and Y takes value in the range -1 and 1. [A.U N/D 2019 (R17) RP]

Solution:

We know that r = 

By Schwavz's inequality,