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Final Exam Solution
ORIE 3500/5500: Engineering Probability and Statistics II
Question 1
(i)
(a) Probability Law: For a statistical experiment with sample space Ω and collection of
events
F
, a probability law is a function
P
:
F →
R
such that
i.
P
(
A
)
≥
0 for any event
A
,
ii.
P
(Ω) = 1, and
iii.
P
(
A
1
∪
A
2
∪
...
) =
P
(
A
1
) +
P
(
A
2
) +
···
for disjoint events
A
1
,A
2
,...
.
(b) Estimator: If
X
1
,...,X
n
are the available data to estimate an unknown parameter
θ
, then a function
g
(
X
1
,...,X
n
) is an estimator for
θ
.
(c) Unbiased estimator: An estimator
ˆ
θ
is unbiased for the parameter
θ
if
E
θ
(
ˆ
θ
) =
θ
∀
θ.
(d) Maximum likelihood estimator: If
X
1
,...,X
n
are the data with joint PDF (or PMF)
L
(
θ
) =
f
X
(
x
1
,...,x
n
,θ
), then the MLE is an estimator
ˆ
θ
which maximizes
L
(
θ
) over
θ
.
(ii)
(a)
M
Z
(
s
) =
E
[
e
s
(
a
(
X

1)+
bY
)
] =
E
[
e
saX
e

sa
e
sbY
] =
e

sa
E
[
e
saX
]
E
[
e
sbY
] =
e

sa
M
X
(
sa
)
M
Y
(
sb
)
.
(b) ˆ
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 Fall '08
 WEBER

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