SYSC4005/5001
Simulation of Discrete Event Systems
Homework Assignment 4
DUE: March 22 2010
1. For the problem of estimating the parameter
θ
of a uniform distribution
in [0
, θ
] we have already seen in class that one unbiased estimator is
W
1
=
n
+ 1
n
Y
max
By following a similar approach as we did in class derive an unbiased
estimator of
θ
based on the minimum observation
Y
min
as
W
2
=
kY
min
.
Find the parameter
k
and derive the relative efficiency of
W
2
with
respect to
W
1
2. If a series of independent Bernoulli trials is terminated with the oc
curence of the first success, the probability function for
K
, the length
of the series, will be given by the geometric distribution
f
K
(
k
;
p
) =
P
(
K
=
k
) = (1

p
)
k

1
p,
k
= 1
,
2
, . . .
where
p
is the (unknown)probability of success at any trial. If
n
such
series are observed, the data will consist of
n
lengths,
k
1
, k
2
, . . . , k
n
.
Find a max. likelihood estimator for the parameter p. Justify in detail
all your arguments!.
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 Winter '10
 LAMBADARIS
 Normal Distribution, Probability theory, Exponential distribution, telephone system, unbiased estimator, systems homework assignment

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