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Rensselaer
Electrical, Computer, and Systems Engineering Department
ECSE 4500 Probability for Engineering Applications
Recitation 1
Wednesday January 27, 2010
Review of Relevant Mathematics
1.
sum of Geometric Series and related
(a) Consider the formula:
∞
X
n
=0
a
n
=
1
1
−
a
Under what conditions on
a
is the formula correct? Why?
(b) More generally, we have, for
n
2
>n
1
:
n
2
X
n
=
n
1
a
n
=
a
n
1
−
a
n
2
+1
1
−
a
,
which holds for any value of
a
,except
a
=1
.
Find this result by using the formula
aS
=
S
+
a
n
2
+1
−
a
n
1
,
where
S
,
P
n
2
n
=
n
1
a
n
.
Show why is this formula true for the geometric series.
(c) Now de
f
ne a generating function as:
G
(
z
)
,
∞
X
n
=0
a
n
z
n
.
1. Show that
G
(1) =
P
∞
n
=0
a
n
=
1
1
−
a
whenever

a

<
1
.
2. Show that
G
0
(1) =
P
∞
n
=0
na
n
=
a
(1
−
a
)
2
whenever

a

<
1
.
This result will be used for
f
nding
the average value of the socalled
geometric
random variable.
3. How can we
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This note was uploaded on 04/15/2010 for the course ECSE 4500 taught by Professor Woods during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
 WOODS

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