STAT 409
Fall 2016
A. Stepanov
Homework #3
(due Friday, September 16, by 4:00 p.m.)
Please include your name ( with your last name underlined ), your NetID,
and your discussion section number at the top of the first page.
No credit will be given without s
STAT 409
Fall 2016
A. Stepanov
Homework #2
(due Monday, September 12, by 4:00 p.m.)
Please include your name ( with your last name underlined ), your NetID,
and your discussion section number at the top of the first page.
No credit will be given without s
STAT 409
1.
Fall 2016
A. Stepanov
Practice Problems 3
Let X 1 , X 2 , , X n be a random sample from the distribution with the probability
density function
f X ( x ; ) = ( + 1 ) (1 x ) ,
~
0 < x < 1,
> 1.
1
2 a consistent estimator of ?
X
a)
Is =
b)
Is =
STAT 409
1 5.
Practice Problems 2
Fall 2016
A. Stepanov
If the random variable Y denotes an individuals income, Paretos law claims
that P ( Y y ) = k , where k is the entire populations minimum income.
y
It follows that
+1
1
f Y ( y ) = k
,
y k;
> 0.
STAT 409
2(r)
The Chi-Square Distribution
f ( y )=
1
(r
yr
2)2 r 2
E(Y) = r
.
Fall 2016
A. Stepanov
Examples for 08/29/2016
2 1 e y 2 ,
0y<
Var ( Y ) = 2 r
Let Z be a N ( 0, 1 ) standard normal random variable.
Then X = Z 2 has a chi-square distribution w
STAT 409
p.m.f. or p.d.f.
1.
Fall 2016
A. Stepanov
Examples for 08/24/2016
f ( x ; ),
.
Suppose = cfw_ 1, 2, 3 and the p.m.f.
parameter space.
f ( x ; ) is
= 1:
f ( 1 ; 1 ) = 0.6,
f ( 2 ; 1 ) = 0.1,
f ( 3 ; 1 ) = 0.1,
f ( 4 ; 1 ) = 0.2.
= 2:
f ( 1 ;
STAT 408
Spring 2016
A. Stepanov
Examples for 6.3
In general, if X 1 , X 2 , , X n is a random sample of size n from a continuous distribution with
cumulative distribution function F ( x ) and probability density function f ( x ), then
F max X ( x ) = P (