Steven Weber
Dept. of ECE
Drexel University
ENGR 361: Statistical Analysis of Engineering Systems (Spring, 2008)
Recitation 9 Solutions (Friday, May 30)
Question 1.
Suppose the population distribution is uniform over
[
a, b
]
:
f
X
(
x
) =
1
b

a
,
a
≤
x
≤
b
0
,
else
.
(1)
Suppose the constants
a, b
are unknown, and hence we gather a random sample,
X
1
, . . . , X
n
to determine them.
Please do the following:
•
Find the ML estimators for
a, b
. Hint: pay attention to the regime where the likelihood function is zero (equiv
alently, where the loglikelihood function is infinitely negative), and that monotonically increasing functions
(derivative of constant sign) are maximized at a boundary point. In particular, form the function:
L
(
x
1
, . . . , x
n
;
a, b
) = log
f
(
x
1
, . . . , x
n
;
a, b
) = log
n
i
=1
f
(
x
i
;
a, b
) =
n
i
=1
log
f
(
x
i
;
a, b
)
,
(2)
noting where
f
(
x
i
;
a, b
) = 0
, and thus where
L
(
x
1
, . . . , x
n
;
a, b
) =
∞
. From here, compute
∂
∂a
L
(
x
1
, . . . , x
n
;
a, b
)
and
∂
∂b
L
(
x
1
, . . . , x
n
;
a, b
)
, and identify the values
a, b
that maximize
L
(
x
1
, . . . , x
n
;
a, b
)
.
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 Spring '04
 Eisenstein
 Standard Deviation, Maximum likelihood, Likelihood function, Monotonic function, Steven Weber

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