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STAT 410
Spring 2008
Version A
Name ANSWERS
.
Quiz 3
(10 points)
Be sure to show all your work, your partial credit might depend on it.
No credit will be given without supporting work.
1.
Let X
1
, X
2
, … , X
n
be a random
sample of size
n
from the distribution
with probability density function
( ) ( )
( )
X
X
1
;
x
x
f
x
f

=
=
,
x
> 1.
a) (5)
Ω
=
{
θ
> 1
}. Find the maximum likelihood estimator
ˆ
of
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Unformatted text preview: θ . Show work. L( θ ) = ( ) ∏ =n i i x 1 & 1 & . ln L( θ ) = ( ) & = ⋅n i i x n 1 ln & 1 & ln . ( ) & == n i i d d x n 1 ln 1 & & & L ln = 0. & = + = n i i x n 1 ln 1 & ˆ . b) (5) Ω = { θ > 2 }. Find the method of moments estimator & ~ of θ . Show work. E(X) = ( ) ( ) 2 & 1 & 1 & 1 & X== ± ± ∞ ∞ ∞⋅ ⋅ dx x x dx x f x . 2 & 1 & 1 1= = & = ⋅ x x n n i i . 1 1 2 & ~= x x ....
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This test prep was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 AlexeiStepanov
 Probability

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