<< Chapter < Page Chapter >> Page >
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 ( 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 ) ) .

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 .

Got questions? Get instant answers now!

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 ( X ) described by

g ( X ) = 200 + 18 I M 1 ( X ) ( X - 10 ) + ( 16 - 18 ) I M 2 ( X ) ( X - 20 )
+ ( 15 - 16 ) I M 3 ( X ) ( X - 30 ) + ( 13 - 15 ) I M 4 ( X ) ( X - 50 )
where M 1 = [ 10 , ) , M 2 = [ 20 , ) , M 3 = [ 30 , ) , M 4 = [ 50 , )

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

Got questions? Get instant answers now!

The problem

If X is a random variable, then Z = g ( X ) is a new random variable. Suppose we have the distribution for X . How can we determine P ( Z 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 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 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 ( g ( X ) M ) .
    1. Mapping approach . Determine the set N of all those t which are mapped into M by the function g . Now if X ( ω ) N , then g ( X ( ω ) ) M , and if g ( X ( ω ) ) M , then X ( ω ) N . Hence
      { ω : g ( X ( ω ) ) M } = { ω : X ( ω ) N }
      Since these are the same event, they must have the same probability. Once N is identified, determine P ( X N ) in the usual manner (see part a, above).
    2. Discrete alternative . For each possible value t i of X , determine whether g ( t i ) meets the defining condition for M . Select those t i which do and add the associated probabilities.

Questions & Answers

short run AC curves?
Jasmin Reply
what is short run AC curves?
Jasmin
what is short run curves?
Jasmin
what is short run curves?
Jasmin
nooo am not from India why!?
Godwin Reply
h
Hamid
Godwin which level of education are you please
Millionaires
millionaires am in SHS 2
Godwin
vnsgu BBA ki first sem ki all subjects ki koi app he ?
Modi Reply
one is business economics app
mudasir
all subjects ki koi nhi he ?
Modi
for accounts u can download accounts complete course app ... step by step Al the topics r mentioned
mudasir
Muje BBA ke first semester ke liye chahiye.... Vnsgu
Modi
there is a app for bba
Muhammad
named as college tutor on play store
Muhammad
isme sab subject ki books aa jayegi ?
Modi
han g
Muhammad
baqi main bad main batata hun abi online paper ho raha ha
Muhammad
okk....
Modi
Apki knsi uni ha
Muhammad
aik aur app ha but wo indian ha
Muhammad
Named as bba books
Muhammad
okk
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
Mera Vnsgu he
Modi
asslam alaikum
Hamid
BBA books app ki photo send kijiye naa please
Modi
photo kesy send ho skty ha yahan sy?
Muhammad
I mean iska option kahan ha
Muhammad
haa.. sorry
Modi
collage tutor mein join teacher wala hi aata he or kuch nhi aata he
Modi
ok
Muhammad
Mera Vnsgu he
Modi
Ap play store py ja ky waha search ker lain wahan sy asani sy ye app mil jay ga apko
Muhammad
ab books kaha lau ?
Modi
oh sorry
Muhammad
main apko sham rak batata hun abi paper ho raha ha☺☺
Muhammad
kya search maru ?
Modi
yar ap ki books ka name kia ha?
Muhammad
modi
Hamid
who know accounting
Hamid
Hmare pass koi bhi subject ki book nhi he
Modi
abi 1st semester ha
Muhammad
ha... First semester
Modi
Yes sir
Modi
mujy book pary ha pdf ma accounting economic statistics Kay book ha mery pass
Hamid
pdf ma h
Hamid
to send karo naa
Modi
ya pa kasy kro
Hamid
kya ?
Modi
whatsp number do
Hamid
number bejo whatsp ka
Hamid
jaldi
Hamid
technical error bta rha he
Modi
number send nhi ho paa rha he
Modi
fb I'd name batio
Hamid
mery Hamid Ali shaikh ha
Hamid
mein nai use krti
Modi
fb I'd ka name bejo
Hamid
aap kaha se ho ?
Modi
fr kiya hoga
Hamid
ap kiya Kiya chlty ho batio
Hamid
mtlb ?
Modi
kiya use krty h wo Batio ma koxhes krta h fr
Hamid
ma pakistan sa h
Hamid
fb dowled Kro ap fr kam hiGa ap ka
Hamid
Krishi Modi iss fb id pe bhejo
Modi
Modi
Muhammad
profile kon sy h
Hamid
Modi are you from india
Muhammad
yr privacy ha Aik I'd pa req nhi jaty
Hamid
who was the father of economic ?why?
Mahesh Reply
who was the known as father of economic?why?
Mahesh
Rationing and hoarding
Semiat Reply
how do the size of a country's population affect labour force
Evans Reply
a mixed economic system
Ngong Reply
What are the types of price elasticity of demand
Juliana Reply
what are massures to promote geographical mobility of labor?
Ngong
Is to make sure that a labourer to know more about his salary to earn before going to the direction
shehu
what is trade
PHILIPPE Reply
Trade is a basic economic concept involving the buying and selling of goods and services, with compensation paid by a buyer to a seller, or the exchange of goods or services between parties. Trade can take place within an economy between producers and consumers.
Miss
what is fisical policy?
ha Reply
fisical policy or fiscal policy?
Miss
what are.the characteristics of economic goods
Hamis
what are the importance of labour market?
Rachael
how discrib the rural development and their four stages
Sheikh Reply
ye economics se related ha
Sheikh
1..traditional stage..no science and technology is applied hence poor productionuu.2..the take off stage..some development strategies are initiated eg transport system is improved but the traditional cultural belief still remain .3..the prematurely stage..technological methods of production are appl
President
applied leading to higher GDP..4..stage of mass consumption..
President
What is Easiest Formula For National Income?
Tenzin Reply
national income/ agrrigate net value
Sheikh
what do you mean by the supply of goods
sachin Reply
supply of good refer to the total unit of production which is ready to sell at a given price
Tenzin
what is implicit cost
fuseini Reply
Yeah
MOHAMED
any cost that has already occurred but not necessarily shown or reported as a separate expense.
President
The links don't seem to be working
Scorch Reply
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask