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Discrete Random Variables and
Their Probability Distribution:
Part I
Cyr Emile M’LAN, Ph.D.
[email protected]
Discrete Random Variable: Part I
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View Full Document Introduction
♠
Text Reference
:
Introduction to Probability and Its
Application, Chapter 4.
♠
Reading Assignment
:
Sections 4.14.2, February
16February 18
In this chapter we extend the concepts and techniques
of probability introduced in Chapter 2.
We’ll also study how probability provides the basis for
making statistical inferences.
Suppose that one flips a coin 100 times and counts the
number of heads. The objective is to infer from the
count that the coin is not balanced.
Discrete Random Variable: Part I
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Introduction
It is reasonable to believe that observing a large
number of heads (say, 90) or a small number of tails
(say, 15) would be an indication of an unbalanced coin.
However, where do one draw the line? At 85 or 75 or 65
or 55?
Without knowing the probability of the frequency
of the number of heads from a balanced coin, one
cannot draw such a line. Hence, we would not be able
to draw any conclusions from the sample of 100 coin
flips.
The concepts and techniques of probability developed
in this chapter will allow us to calculate such thresholds
we seek.
Discrete Random Variable: Part I
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View Full Document Random Variables
Sample space need not consist of numbers. But very
often we are interested in numerical outcomes
associated with the random phenomenon generating
the sample space.
Example 4.1
:
Consider an experiment where we flip two balanced
coins and observe the results.
The sample space is
S
=
{
(
H,H
)
,
(
H,T
)
,
(
T,H
)
,
(
T,T
)
}
.
However, one can list the events in a different way by
counting the number of heads (or if we wish, the
number of tails). Thus, the four simple events are
replaced now by: 2 heads, 1 head, 1 head, 0 heads,
resp.
The number of heads,
X
, is called a
random variable
(r.v.). The induced sample space associated to the r.v.
X
is
S
s
=
{
0
,
1
,
2
}
.
Discrete Random Variable: Part I
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Random Variables
Example 4.2
:
In the game of craps played in casinos, the player
tosses two dice. A natural way of listing the events is to
describe the number on the first die and the number on
the second die. Hence, the sample space is.
S
=
{
(
i,j
)
,i,j
= 1
,
2
,
3
,
4
,
5
,
6
}
.
However, in this game the player is primarily interested
in the total sum of the two dice. If we define the random
variable
X
to be the total of the two dice,
X
can take
integer values between 2 and 12. Hence, the induced
sample space for
Y
is
S
s
=
{
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
}
.
Discrete Random Variable: Part I
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Definition 4.1
:
A
random variable
is a realvalued function or rule defined
over a sample space that assigns a number to each outcome
of a random experiment.
Types of Random Variables
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This document was uploaded on 11/18/2010.
 Spring '09
 Probability

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