Two dimensional random variables

UNIT - II

TWO DIMENSIONAL RANDOM VARIABLES

Joint distributions Marginal and conditional distributions - Covariance - Correlation and Linear Regression - Transformation of random variables. Central limit theorem (for independent and identically distributed random variables)

Introduction

In unit 1, various aspects of the theory of single random variable were studied. The random variable was found to be a powerful concept. It enabled many realistic problems to be described in a probabilistic way such that practical measures could be applied to the problem even though it was random. From knowledge of the probability distribution or density function of impact position, we can solve for such practical measures as the mean value of impact position, its variance, and skew. These measures are not, however, a complete enough description of the problem in most cases. It may be necessary to extend the theory to include several random variables. Fortunately, many situations of our engineering problems are handled by the theory of two random variables. Hence, such important concepts as auto correlation, cross-correlation and covariance functions, which apply to random processes, are based on two random variables.

1. Two-dimensional random variable: Let S be the sample space. Let X = X (s) and Y = Y (s) be two functions each assigning a real number to each outcome s E S. Then (X, Y) is a two-dimensional random variable.

2. Two-dimensional discrete random variables: If the possible values of (X, Y) are finite or countably infinite, then (X, Y) is called a two-dimensional discrete random variable. When (X, Y) is a two-dimensional discrete random variable the possible values of (X, Y) may be represented as (xi, yj), i = 1, 2, ... n, j = 1, 2, 3, ... m.

3. Two-dimensional continuous random variables: If (X, Y) can assume all values in a specified region R in the XY plane (X, Y) is called a two-dimensional continuous random variable.