Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
&
Computer
Science
6.041/6.431:
Probabilistic
Systems
Analysis
(Fall
2010)
Problem
Set
7
Due
November
8,
2010
1. Consider
a
sequence of
mutually independent,
identically distributed,
probabilistic
trials.
Any
particular trial results in either
a
success (with probability
p
) or
a failure.
(a)
Obtain a
simple expression for the
probability that the
i
th success occurs before
the
j
th
failure. You may leave your
answer
in the
form
of a summation.
(b)
Determine the expected value and variance
of the
number
of successes which occur
before
the
j
th failure.
(c)
Let
L
17
be described by a
Pascal
PMF of order
17.
Find the
numerical
values of
a
and
b
in
the following
equation. Explain
your
work.
∞
a
�
�
�
�
b
p
L
17
(
l
) =
p
x
(1
−
p
)
(
b
−
x
)
x
l
=42
x
=0
2. Fred is giving
out
samples of dog
food.
He
makes calls door to door, but he leaves a sample (one
can)
only
on those calls for
which the
door
is answered
and
a dog is in residence.
On any call
the probability of
the door
being
answered is 3
/
4,
and the
probability that any household
has
a dog is 2
/
3. Assume that
the events “Door
answered” and “A
dog lives here” are
independent
and also that
the outcomes of
all calls are
independent.
(a)
Determine the probability that
Fred gives away his first sample
on his third call.
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 Fall '10
 Prof.DimitriBertsekas
 Electrical Engineering, Probability, Probability theory, Probability space, Probabilistic Systems Analysis, numb er

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