# Functions of a random variable

 Page 1 / 3
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.

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive