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lecture 8,
Discrete
Random
Variables I
1/24
Random
Variable
Discrete
Probability
Distributions
Expected
Value
Linearity of
Expectations
Variance
lecture 8, Discrete Random Variables I
Outline
1
Random Variable
2
Discrete Probability Distributions
3
Expected Value
Linearity of Expectations
4
Variance
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Discrete
Random
Variables I
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Random
Variable
Discrete
Probability
Distributions
Expected
Value
Linearity of
Expectations
Variance
lecture 8, Discrete Random Variables I
Random Variable
Random Variable
±
A
random variable
is a function, say
X
, from a sample
space
S
to the real numbers
R
:
•
Classiﬁcation of random variable: A random variable is
said to be
•
discrete
if it takes on ﬁnitely or countably many values
•
E.g.: number of heads when tossing a coin 4 times
•
continuous
if it may assume any value in one or more
intervals
•
E.g.: ﬁnish time of the winning horse in a horse racing
•
Focus on discrete random variable in this chapter
•
Convention: use capital letters for random variables,
and lowercase letters for the possible values
lecture 8,
Discrete
Random
Variables I
3/24
Random
Variable
Discrete
Probability
Distributions
Expected
Value
Linearity of
Expectations
Variance
lecture 8, Discrete Random Variables I
Random Variable
Example: Coin tossing
•
Toss a (fair) coin 4 times.
•
Sample space = {HHHH, HHHT, HHTH, HHTT, HTHH,
HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT,
TTHH, TTHT, TTTH, TTTT }
•
All are equally like, hence each outcome has probability
1/16
•
X
:= # heads
•
E.g.,
X
(
HHHH
) =
4
,
X
(
HHHT
) =
3
, etc.
•
The possible values that
X
can take are
,
hence
X
is a discrete
random variable
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Discrete
Random
Variables I
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Random
Variable
Discrete
Probability
Distributions
Expected
Value
Linearity of
Expectations
Variance
lecture 8, Discrete Random Variables I
Discrete Probability Distributions
Discrete Probability Distributions
±
Let
X
be a discrete random variable. The
probability
distribution
of
X
identiﬁes the probability associated
with each possible value that
X
can take.
•
The probability distribution can be represented by a
table, or a graph, or a formula.
•
For each value
x
that
X
can take, the probability
P
(
X
=
x
)
is denoted by
p
(
x
)
lecture 8,
Discrete
Random
Variables I
5/24
Random
Variable
Discrete
Probability
Distributions
Expected
Value
Linearity of
Expectations
Variance
lecture 8, Discrete Random Variables I
Discrete Probability Distributions
Example: Coin tossing,
ctd
•
X
:= # heads when tossing a (fair) coin 4 times.
•
What’s the probability distribution of
X
?
•
The possible values
X
can take are
{
0
,
1
,...,
4
}
•
Need to ﬁnd for each
x
=
0
,
1
...,
4, the probability
P
(
X
=
x
)
•
{
X
=
0
}
: consists of one outcome: TTTT, hence
P
(
X
=
0
)
=1/16
•
{
X
=
1
}
: consists of 4 outcome: HTTT, THTT, TTHT,
TTTH, hence
P
(
X
=
1
)
=4/16
•
Similarly,
P
(
X
=
2
)
= 6/16,
P
(
X
=
3
) =
4
/
16, and
P
(
X
=
4
) =
1
/
16
•
Hence the probability distribution of
X
can be
represented by the following table
x
0
1
2
3
4
P
(
X
=
x
)
1/16
4/16
6/16
4/16
1/16
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Discrete
Random
Variables I
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Random
Variable
Discrete
Probability
Distributions
Expected
Value
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This note was uploaded on 12/28/2010 for the course BUSINESS A ISOM 111 taught by Professor Yingyingli during the Fall '10 term at HKUST.
 Fall '10
 YingYingLi
 Business

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