1
Lecture 17 Exercises
1.
Suppose in a sequence of Bernouilli trials, we observe
12
,
,...,
n
y
yy
where
(
1)
i
PY
!
""
,
(
0)
1
i
"
" #
a)
Show that the probability function for
i
Y
can be written
1
( )
(1
)
,
0,1
p y
y
!!
#
"
#
"
We have
0
1
1
0
(0)
)
1
,
(0)
)
pp
"
#
" #
"
#
"
as required.
b)
Construct the likelihood function
()
L
and the log-likelihood function
l
The likelihood function is the probability of the observed data, as a function of
.
11
1
1
2
2
1
1
( )
(
)
(
) ...
(
) (using independence)
)
...
)
)
nn
ii
y
n
y
L
P Y
y
P Y
y
P Y
y
#
#
#
"
"
$
"
$ $
"
"
#
$ $
#
%%
"#
and the log-likelihood function is
( )
ln ( )
ln( )
(
)ln(1
)
l
L
y
n
y
"
"
&
#
#
%
%
c)
Find the MLE
ˆ
. Be sure to show that it is a maximum.
We have
2
2
2
2
( )
( )
,
1
)
i
i
i
i
y
n
y
y
n
y
dl
d l
dd
##
"
#
" #
#
%
%
%
%
so to find the MLE, set
0
dl
d
"
and solve. We have
0
ˆˆ
1
y
n
y
#
#"
#
or
ˆ
i
y
n
"
%
. Since the second derivative at
ˆ
is
negative, we know that we have found a maximum of the log likelihood.
d)
Show that
( ,
,...,
)
n
y y
y
"
is a sufficient statistic.
We can express
l
as
( )
[ ln( ) (1
)ln(1
)]
ln
"
& #
#
so that
l
depends on
,
,...,
n
y
y
y only through
ˆ
. Hence
( ,
,...,
)
n
y y
y
"
is a sufficient statistic.
e)
Find
[ ]
E
!
and
[ ]
Var
!
.
We have
1
n
i
i
Y
Y
"
%
!
where
~
( , )
Y
binom n
. Hence
[ ]
[ ]
)
[ ]
,
[ ]
E Y
Var Y
E
Var
n
n
n
#
"
"
"
"
.
f)
Suppose
20
1
20,
12
i
i
ny
"
%
. What is the relative likelihood function
( )
R
?