Statistics 621: Homework #9. Solution
Due Wednesday, March 9
1. (2 pts)
(i)
k
k
n xi
n
(1 )nxi =
xi
xi
i=1
f (xk ) =
i=1
() =
k
i=1
xi
k
i=1
(1 )kn
1
1 (1 ) 1 ;
B (, )
xi
,
with square error loss.
> 0, > 0, 0 < < 1
The Bayes estimation is an estimation
Statistics 621: Homework #8. Solution
Due Friday, March 4
1. (1.5 pts)
X N (, 1); L(, a) = ( a)2 ; Note E [ f (X )]2 = V ar(f (X ) + [E (f (X ) ]2 = V ar(f (X ) + bias2
(i) For 1 (x) = x,
R(, 1 (X ) = E [L(, 1 (X )] = E ( X )2 = V ar(X ) = 1
(ii) For 2 (x
Statistics 621: Homework #7. Solution
Due Friday, February 25
(a) (2 pts)
E, (X1 )
ex
dx,
x )2
(1 + e
log(t 1) 1
dt, where t = 1 + ex
t2
1
1
1 log(t 1)
dt
dt,
1 t2
1
t2
0, using integration by parts
.
=
x
=
=
=
=
2
E, X1
x2
=
=
=
=
=
=
ex
dx,
(1 + e
Statistics 621: Homework #6. Solution
Due Friday, February 11
1.
(a) (1 pts) Let W denote U + (V U ) which is any UE of .
Then, varW =varU + 2cov(U, V U ) + 2 var(V U ).
If cov(U, V U ) = a > 0, then varW < varU when [2a + var(V U )] < 0, i.e., for var(2a
Statistics 621: Homework #5. Solution
Due Friday, February 4
1.(2 pts)
E (X )
k
k+r1 r
p (1 p)k ,
k
k
=
(k + r 1)! r
p (1 p)k ,
k !(r 1)!
k=0
=
k=1
=
k=1
=
(k + r 1)! r
r
p (1 p)k ,
(k 1)!(r 1)!
r
r(1 p)
p
=
r(1 p)
p
=
k=1
k + r 1 r+1
p (1 p)k1 ;
k1
set l
Statistics 621: Homework #4. Solution
Due Friday, January 28
1.(a)(1 pts)
n
L(, x)
=
(xi )1
n
i=1
n
log L(, x)
=
n log + ( 1)
log xi
i=1
n
d
log L(, x)
d
n
set
+
log xi = 0 =
i=1
d2
log L(, x)
d2
Thus, =
=
=
n
log Xi
n
i=1
n
< 0,
2
1
log xi
n
i=1
> 0
Statistics 621: Homework #3. Solution
Due Friday, January 21
1. (1.5 pts)
We want to show the distribution of (X1 , X2 , ., Xn ) given Sn = t is independent of .
P (X1 = x1 , X2 = x2 , ., Xn = xn | Sn = t),
P (X1 = x1 , X2 = x2 , ., Xn = xn , n xi = t)
i=
Statistics 621: Homework #2. Solution
Due Friday, January 14
1. (1.5 pts)
The joint pmf of X s is
P (X = x) =
Thus,
n
i=1 Xi
n
1
1
e
n
i=1
n
xi
i=1
1
xi !
is a sucient statistic for by the factorization theorem.
2. (2 pts)
Dene an indicator variable,
1, i
Statistics 621: Homework #1. Solution
Due Friday, January 7
1. Four identical, sealed envelopes are on a table. One contains a $100 bill and the others
are empty.
You select an envelope at random and hold it in your hand without opening it.
Two of the t