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Binomial and Normal Random Variables
Charles B. Moss
July 14, 2010
I. Bernoulli Random Variables
A. The Bernoulli distribution characterizes the coin toss. Specifcally,
there are two events
X
=0
,
1 with
X
= 1 occurring with probabil
ity
p
. The probability distribution Function
P
[
X
] can be written
as
P
[
X
]=
p
x
(1
−
p
)
x
(1)
B. Next, we need to develop the probability oF where both and are
identically distributed. IF the two events are independent, the
probability becomes
P
[
X,Y
]=
P
[
X
]
P
[
Y
]=
p
x
p
y
(1
−
p
)
1

x
(1
−
p
)
1

y
=
p
x
+
y
(1
−
p
)
2

x

y
(2)
C. Now, this density Function is only concerned with three outcomes
Z
=
X
+
Y
=
{
0
,
1
,
2
}
.
1. There is only one way each For
Z
=0or
Z
=2
.
a) Specifcally For
Z
=0,
X
=0and
Y
=0
.
b) Similarly, For
Z
=2,
X
=1and
Y
=1
.
2. However, For
Z
= 1 either
X
=1and
Y
=0or
X
=0and
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View Full Document AEB 6571 Econometric Methods I
Professor Charles B. Moss
Lecture XI
Fall 2010
P
[
Z
=0
]=
p
0
(1
−
p
)
2

0
P
[
Z
=1
]=
P
[
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
 Spring '10
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