Functions of a random variable

Introduction

Frequently, we observe a value of some random variable, but are really interested in a value derived from this by a function rule. If X is a random variable and g is a reasonable function (technically, a Borel function ), then $Z=g\left(X\right)$ is a new random variable which has the value $g\left(t\right)$ for any ω such that $X\left(\omega \right)=t$ . Thus $Z\left(\omega \right)=g\left(X\right(\omega \left)\right)$ .

The problem; an approach

We consider, first, functions of a single random variable. A wide variety of functions are utilized in practice.

The problem

If X is a random variable, then $Z=g\left(X\right)$ is a new random variable. Suppose we have the distribution for X . How can we determine $P(Z\in M)$ , the probability Z takes a value in the set M ?

An approach to a solution

We consider two equivalent approaches

1. To find $P(X\in M)$ .
   1. Mapping approach . Simply find the amount of probability mass mapped into the set M by the random variable X .
      • In the absolutely continuous case, calculate ${\displaystyle {\int }_M{f}_X}$ .
      • In the discrete case, identify those values t _i of X which are in the set M and add the associated probabilities.
   2. Discrete alternative . Consider each value t _i of X . Select those which meet the defining conditions for M and add the associated probabilities. This is the approach we use in the MATLAB calculations. Note that it isnot necessary to describe geometrically the set M ; merely use the defining conditions.
2. To find $P\left(g\right(X)\in M)$ .
   1. Mapping approach . Determine the set N of all those t which are mapped into M by the function g . Now if $X\left(\omega \right)\in N$ , then $g\left(X\right(\omega \left)\right)\in M$ , and if $g\left(X\right(\omega \left)\right)\in M$ , then $X\left(\omega \right)\in N$ . Hence
      $$\{\omega :g(X\left(\omega \right))\in M\}=\{\omega :X(\omega )\in N\}$$
      Since these are the same event, they must have the same probability. Once N is identified, determine $P(X\in N)$ in the usual manner (see part a, above).
   2. Discrete alternative . For each possible value t _i of X , determine whether $g\left(t_i\right)$ meets the defining condition for M . Select those t _i which do and add the associated probabilities.