Two Dimensional Random Variables

TWO DIMENSIONAL RANDOM VARIABLES

 

·        Introduction

·        Joint distribution

·        Marginal and Conditional Distribution

·        Covariance

·        Correlation Coefficient

·        Linear Regression

·        Transformation of random variables

 

Introduction

 

In the previous chapter we studied various aspects of the theory of a single R.V. In this chapter we extend our theory to include two R.V's one for each coordinator axis X and Y of the XY Plane.

 

DEFINITION : Let S be the sample space. Let X = X(S) & Y = Y(S) be two functions each assigning a real number to each outcome s ∈ S. hen (X, Y) is a two dimensional random variable.

 

1 Types of random variables

1.     Discrete R.V.’s

2.     Continuous R.V.’s

Discrete R.V.’s (Two Dimensional Discrete R.V.’s)

If the possible values of (X, Y) are finite, then (X, Y) is called a two dimensional discrete R.V. and it can be represented by (xi, y), i = 1,2,….,m.

In the study of two dimensional discrete R.V.’s we have the following 5 important terms.

 

•      Joint Probability Function (JPF) (or) Joint Probability Mass Function.

 

•      Joint Probability Distribution.

 

•      Marginal Probability Function of X.

 

•      Marginal Probability Function of Y.

 

•      Conditional Probability Function.

 

1.1 Joint Probability Function of discrete R.V.’s X and Y

The function P(X = x_i, Y = y_j) = P(x_i, y_j) is called the joint probability function for discrete random variable X and Y is denote by pij.

Note

 

1.     P(X = x_i, Y = y_j) = P[(X = x_i)∩(Y = y_j)] = p_ij

2.     It should satisfies the following conditions

(i) p_ij ≥ ∀ i, j          (ii) Σ_jΣ_i p_ij = 1

 

1.2 Marginal Probability Function of X

If the joint probability distribution of two random variables X and Y is given then the marginal probability function of X is given by

P_x(x_i) = p_i    (marginal probability function of Y)

 

Conditional Probabilities

The conditional probabilities function of X given Y = y_j is given by

 

 

SOLVED PROBLEMS ON MARGINAL DISTRIBUTION

Example:2.1.1

 

From the following joint distribution of X and Y find the marginal distributions.

 

 

3 CONTINUOUS RANDOM VARIABLES

 

•   Two dimensional continuous R.V.’s

If (X, Y) can take all the values in a region R in the XY plans then (X, Y) is called two-dimensional continuous random variable.

 

•             Joint probability density function :

 

 

4 REGRESSION

* Line of regression

 

The line of regression of X on Y is given by

 

 

Example:2.4.1

1. From the following data, find

 

(i) The two regression equation

 

(ii) The coefficient of correlation between the marks in Economic and Statistics.

 

(iii) The most likely marks in statistics when marks in Economic are 30.

 

 

 

5 COVARIANCE

Def : If X and Y are random variables, then Covariance between X and Y is defined as

Cov (X, Y)  = E(XY) – E(X) . E(Y)

Cov (X, Y)  = 0    [If X & Y are independent]

 

 

6 CORRELATION

Types of Correlation

 

•       Positive Correlation

(If two variables deviate in same direction)

 

•       Negative Correlation

(If two variables constantly deviate in opposite direction)

 

7 KARL-PEARSON’S COEFFICIENT OF CORRELATION

Correlation coefficient between two random variables X and Y usually denoted by r(X, Y) is a numerical measure of linear relationship between them and is defined as

 

SOLVED PROBLEMS ON CORRELATION

Example :2.6.1

Calculated the correlation coefficient for the following heights of fathers X and their sons Y

 

 

8 TRANSFORMS OF TWO DIMENSIONAL RANDOM VARIABLE

Formula:

 

 

 

TUTORIAL QUESTIONS

 

1.     The jpdf of r.v X and Y is given by f(x,y)=3(x+y),0<x<1,0<y<1,x+y<1 and 0 otherwise. Find the marginal pdf of X and Y and ii) Cov(X,Y).

2.     Obtain the correlation coefficient for the following data:

 

X:	68	64	75	50	64	80	75	40	55	64
Y:	62	58	68	45	81	60	48	48	50	70

3.The two lines of regression are 8X-10Y+66=0, 4X-18Y-214=0.The variance of x is 9 find i) The mean value of x and y. ii) Correlation coefficient between x and y.

4. If X1,X2,…Xn are Poisson variates with parameter λ=2, use the central limit theorem to find P(120≤Sn≤160) where Sn=X1+X2+…Xn and n=75.

5. If the joint probability density function of a two dimensional random variable (X,Y) is

 

given by f(x, y) = x² +                                                                                     , 0<x<1,0<y<2= 0, elsewhere Find (i) P(X>1/2)(ii) P(Y<X) and                                                   (iii)

P(Y<1/2/ X<1/2).

6. Two random variables X and Y have joint density Find Cov (X,Y).

 

7. If the equations of the two lines of regression of y on x and x on y are respectively 7x-16y+9=0; 5y-4x-3=0, calculate the coefficient of correlation.

 

 WORKEDOUT EXAMPLES

 

Example 1

The j.d.f of the random variables X and Y is given

 f (x, y) =          8xy, 0 <x <1, 0 y< x  

  otherwise        0,      

        

Find (i) fX(x)  (ii) fY(y)  (iii) f(y/x)  

Solution  

We know that        

(i)                The marginal pdf of ‘X’ is