10.1 Function of random vectors  (Page 13/6)

Introduction

The general mapping approach for a single random variable and the discrete alternative extends to functions of more than one variable. It is convenient to consider thecase of two random variables, considered jointly. Extensions to more than two random variables are made similarly, although the details are more complicated.

The general approach extended to a pair

Consider a pair $\{X,Y\}$ having joint distribution on the plane. The approach is analogous to that for a single random variable with distribution on the line.

1. To find $P\left(\right(X,Y)\in Q)$ .
   1. Mapping approach . Simply find the amount of probability mass mapped into the set Q on the plane by the random vector $W=(X,Y)$ .
      • In the absolutely continuous case, calculate ${\displaystyle \int {\int }_Q{f}_{XY}}$ .
      • In the discrete case, identify those vector values $(t_i,u_j)$ of $(X,Y)$ which are in the set Q and add the associated probabilities.
   2. Discrete alternative . Consider each vector value $(t_i,u_j)$ of $(X,Y)$ . Select those which meet the defining conditions for Q and add the associated probabilities. This is the approach we use in the MATLAB calculations. It does not require that we describe geometricallythe region Q .
2. To find $P\left(g\right(X,Y)\in M)$ . g is real valued and M is a subset the real line.
   1. Mapping approach . Determine the set Q of all those $(t,u)$ which are mapped into M by the function g . Now
      $$W\left(\omega \right)=\left(X\right(\omega ),Y(\omega \left)\right)\in Q\text{iff}g\left(\right(X\left(\omega \right),Y\left(\omega \right))\in M\text{Hence}$$
      $$\{\omega :g(X\left(\omega \right),Y\left(\omega \right))\in M\}=\{\omega :(X\left(\omega \right),Y\left(\omega \right))\in Q\}$$
      Since these are the same event, they must have the same probability. Once Q is identified on the plane,determine $P\left(\right(X,Y)\in Q)$ in the usual manner (see part a, above).
   2. Discrete alternative . For each possible vector value $(t_i,u_j)$ of $(X,Y)$ , determine whether $g(t_i,u_j)$ meets the defining condition for M . Select those $(t_i,u_j)$ which do and add the associated probabilities.

We illustrate the mapping approach in the absolutely continuous case. A key element in the approach is finding the set Q on the plane such that $g(X,Y)\in M$ iff $(X,Y)\in Q$ . The desired probability is obtained by integrating $f_{XY}$ over Q .