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Final Examination
Ramesh Johari
March 20, 2009
Instructions
1. Take alternate seating.
2. Answer all questions in the blue examination books. Answers given on any other paper will
not be counted.
3. The examination begins at 8:30 am, and ends at 11:30 am.
4. No notes, books, calculators, or other aids are allowed.
5. Show your work! Partial credit will be given for correct reasoning.
Honor Code
In taking this examination, I acknowledge and accept the Stanford University Honor Code.
NAME
(signed)
NAME
(printed)
1
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View Full DocumentUseful Formulas
1. If
X
is a geometric random variable with parameter
p
, then the pmf of
X
is:
P
(
X
=
k
) = (1

p
)
k

1
p, k
≥
1
.
2. If
X
is a binomial random variable with parameters
p
and
n
, then the pmf of
X
is:
P
(
X
=
k
) =
p
n
k
P
p
k
(1

p
)
n

k
=
n
!
k
!(
n

k
)!
p
k
(1

p
)
n

k
,
0
≤
k
≤
n.
3. If
T
is an exponentially distributed random variable with mean
1
/λ
, then the density of
T
is
given by
f
T
, where:
f
T
(
t
) =
λe

λt
,
t
≥
0
.
4. If
S
is a random variable with a gamma distribution of parameters
n
and
λ
, where
n
is a
positive integer, then the density of
S
is given by
f
S
, where:
f
S
(
s
) =
λe

λs
(
λs
)
n

1
(
n

1)!
,
s
≥
0
.
5. If
N
is a Poisson random variable of parameter
λ
, then the pmf of
N
is:
P
(
N
=
k
) =
e

λ
λ
k
k
!
, k
≥
0
.
6. For an
M/M/
1
queue with arrival rate
λ
and service rate
μ
with
λ < μ
, the equilibrium
distribution is:
P
(
Q
=
j
) = (1

ρ
)
ρ
j
,
j
≥
0
,
where
ρ
=
λ/μ
.
7. For an
M/M/
∞
queue with arrival rate
λ
and service rate
μ
, the equilibrium distribution is:
P
(
Q
=
j
) =
e

ρ
ρ
j
j
!
,
j
≥
0
,
where
ρ
=
λ/μ
.
8. For an
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 Winter '11
 Ramesh

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