ECSE 305 Lecture Notes - Fair Coin, Random Variable, Inverse Function

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Published on 18 Apr 2013
School
McGill University
Department
Electrical Engineering
Course
ECSE 305
Professor
Chapter 5
Introduction to Random Variables
Consider a random experiment described by a triplet (S, F, P ). In applica-
tions of probabilities, we are often interested in numerical quantities derived
from the experimental outcomes. These quantities may be viewed as func-
tions from the sample space Sinto the set of real numbers R, as in:
sSX(s)R
Provided certain basic requirements are satisfied, these quantities are gener-
ally called random variables.
As an example, consider the sum of the two numbers showing up when rolling
a fair die twice:
The set of all possible outcomes is S={(i, j) : i, j ∈ {1,2, ..., 6}}.
The sum of the two numbers showing up may be represented by the
functional relationship
s= (i, j)X(s) = i+j.
Note that the function X(s) may be used in turn to define more complex
events. For instance, the event that the sum is greater or equal to 11
may be expressed concisely as A={sS:X(s)11}
117
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118
The terminology random variable is appropriate in this type of situations
because:
The value X(s) depends on the experimental outcome s.
The outcome sof a particular realization of the random experiment
(i.e. a trial) is unknown beforehand, and so is X(s).
Each experimental trial may lead to a different value of X(s)
Random variables are extremely important in engineering applications. They
are often used to model physical quantities of interest that cannot be pre-
dicted exactly due to uncertainties. Some examples include:
Voltage and current measurements in an electronic circuit.
Number of erroneous bits per second in a digital transmission.
Instantaneous background noise amplitude at the output of an audio
amplifier.
Modelization of such quantities as random variables allows the use of proba-
bility in the design and analysis of these systems.
This and the next few Chapters are devoted to the study of random variables,
including: definition, characterization, standard models, properties, and a lot
more...
In this Chapter, we give a formal definition of a random variable, we introduce
the concept of a cumulative distribution function and we introduce the basic
types of random variables.
c
2003 Benoˆıt Champagne Compiled February 10, 2012
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5.1 Preliminary notions 119
5.1 Preliminary notions
Function from Sinto R:
Let Sdenote a sample space of interest.
A function from Sinto Ris a mapping, say X, that associate to every
outcome in Sa unique real number X(s):
S
mapping X
real axis
s
1
s
2
s
3
X(s
1
) X(s
2
)=X(s
3
)
Figure 5.1: Illustration of a mapping Xfrom Sinto R.
The following notation is often used to convey this idea:
X:sSX(s)R.(5.1)
We refer to the sample space Sas the domain of the function X.
The range of the function X, denoted RX, is defined as
RX={X(s) : sS} ⊆ R(5.2)
That is, RXis the of all possible values for X(s), or equivalently, the
set of all real numbers that can be “reached” by the mapping X.
c
2003 Benoˆıt Champagne Compiled February 10, 2012
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Document Summary

Consider a random experiment described by a triplet (s,f, p ). In applica- tions of probabilities, we are often interested in numerical quantities derived from the experimental outcomes. These quantities may be viewed as func- tions from the sample space s into the set of real numbers r, as in: s s x(s) r. Provided certain basic requirements are satis ed, these quantities are gener- ally called random variables. For instance, the event that the sum is greater or equal to 11 may be expressed concisely as a = {s s : x(s) 11} Random variables are extremely important in engineering applications. They are often used to model physical quantities of interest that cannot be pre- dicted exactly due to uncertainties. Some examples include: voltage and current measurements in an electronic circuit, number of erroneous bits per second in a digital transmission, instantaneous background noise amplitude at the output of an audio ampli er.

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