Functions of a random variable

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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(X)is a new random variable which has the value g(t) for any ω such that X(ω)=t. Thus Z(ω)=g(X(ω)). Suppose we have the distribution for X. How can we determine P(Z∈M), the probability Z takes a value in the set M? Mapping approach. Simply find the amount of probability mass mapped into the set M by the random variable X. In the absolutely continuous case, integrate the density function for X over the set M. In the discrete case, as an alternative, select those possible values for X which are in the set M and add their probabilities.For a Borel function g and set M, determine the set N of all those t which are mapped into M, then determine the probability X is in N as in the previous case.

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 $Z\left(\omega \right)=g\left(X\left(\omega \right)\right)$ .

The problem; an approach

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

A quality control problem

In a quality control check on a production line for ball bearings it may be easier to weigh the balls than measure the diameters. If we can assume true spherical shape and w is the weight, then diameter is $k{w}^{1/3}$ , where k is a factor depending upon the formula for the volume of a sphere, the units of measurement, and the density of the steel.Thus, if X is the weight of the sampled ball, the desired random variable is $D=k{X}^{1/3}$ .

Price breaks

The cultural committee of a student organization has arranged a special deal for tickets to a concert. The agreement is that the organization will purchase ten ticketsat $20 each (regardless of the number of individual buyers). Additional tickets are available according to the following schedule: • 11-20,$18 each
• 21-30, $16 each • 31-50,$15 each
• 51-100, \$13 each

If the number of purchasers is a random variable X , the total cost (in dollars) is a random quantity $Z=g\left(X\right)$ described by

$g\left(X\right)=200+18{I}_{M1}\left(X\right)\left(X-10\right)+\left(16-18\right){I}_{M2}\left(X\right)\left(X-20\right)$
$+\left(15-16\right){I}_{M3}\left(X\right)\left(X-30\right)+\left(13-15\right){I}_{M4}\left(X\right)\left(X-50\right)$
$\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M1=\left[10,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M2=\left[20,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M3=\left[30,\phantom{\rule{0.166667em}{0ex}}\infty \right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M4=\left[50,\phantom{\rule{0.166667em}{0ex}}\infty \right)$

The function rule is more complicated than in [link] , but the essential problem is the same.

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 $P\left(Z\in M\right)$ , the probability Z takes a value in the set M ?

An approach to a solution

We consider two equivalent approaches

1. To find $P\left(X\in M\right)$ .
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 ${\int }_{M}{f}_{X}$ .
• 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 $P\left(g\left(X\right)\in M\right)$ .
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 $g\left(X\left(\omega \right)\right)\in M$ , and if $g\left(X\left(\omega \right)\right)\in M$ , then $X\left(\omega \right)\in N$ . Hence
$\left\{\omega :g\left(X\left(\omega \right)\right)\in M\right\}=\left\{\omega :X\left(\omega \right)\in N\right\}$
Since these are the same event, they must have the same probability. Once N is identified, determine $P\left(X\in N\right)$ 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.

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