1.
The concentration of cadmium in a lake is measured 17 times. The
measurements average 211 parts per billion with an SD of 15 parts per billion.
Could the real concentration of cadmium be below the standard of 200 ppb?.
This is a standard hypothesis tes
1.
Page 362, Q 2.
The log-likelihood is
which must be maximized subject to
Lagrange multipliers to see that
where
is the Lagrange multiplier. Since
that
2.
. Use
we find
so
.
Page 362, Q 3.
Now the log-likehood is
which yields the likelihood equations
and
STAT 801
Assignment 8: Solutions
1.
Suppose
level
are independent Poisson( ) variables. Find the UMP
test of
case n=3 and
versus
and evaluate the constants for the
.
The likelihood ratio for 1 versus
is large or equivalently if
rejects if
is large. To get
Midterm 1: Solutions
1.
Suppose that X and Y are independent and that each has density, f, given
by
for t>0 and
1.
for t<0.
Find the joint density of
and V=X+Y. [4 marks]
Solving for X and Y we get
The Jacobian is then
2.
and
.
and the joint density is
Fi
STAT 801
Solution Bits: Assignment 6
1.
Page 365. Q 21.
The mean of a Uniform
distribution is
find that
1.
is unbiased so is
. Since
equal to its variance which is
is
2.
make
all
is then
.
provided that all
's are less than . To
's less than is
is
The bia
STAT 801 Solutions: Assignment 5
1.
Page 362, Q 2.
The log-likelihood is
which must be maximized subject to
Lagrange multipliers to see that
where
is the Lagrange multiplier. Since
that
2.
. Use
we find
so
.
Page 362, Q 3.
Now the log-likehood is
which yi
STAT 801
Solutions: Assignment 4
1.
page 213, number 7
To compute
independent. Now
you can compute
because X and Y are
and similarly for Y.
To compute the distribution of XY define U=XY and V=Y, say. Then you
should draw a picture to see what possible val
STAT 801
Solutions: Assignment 3
1.
Number 37 on page 216.
1.
For
the cdf of Y is clearly 0. For
the cdf is evidently 1.
For
(The cdf of X is trivial to find.) Differentiating we get that the density
of Y is
2.
The first two lines above are the same but n
STAT 801 Solutions: Assignment 7
1.
Page 398. Q 1.
One pivot is
which is Uniform on [0,1]. So
which leads to the interval
The event
can be rewritten as
whose probability is
Take an interval of the form aY,bY whose coverage probability is
and minimize b-a
STAT 801
Solutions: Asst 2
1.
Suppose
Let
1.
are iid real random variables with density f .
be the X 's arranged in increasing order.
Find the joint density of
.
Let g be the joint density and
.
Since
we may take g to be 0 on
For
Divide by
in A choose so
Density
8
200
220
240
260
280
300
320
340
160
120
60 80
Birth Weight (ounces)
Gestation period (days)
200
220
240
260
280
300
Gestation period (days)
320
340
A
Ask Richard if the wierd typo (caused by the % sign in LTEX) caused visible diculty
for the stu
Sampling designs leading to chi-squared:
1) Several samples, say one in each column of
table.
Each sampled unit classied into one row.
Jargon: one margin xed.
2) One sample.
Each sampled unit classied in two ways: rows
and columns.
Jargon: neither margin
1.
Suppose
level
are independent Poisson( ) variables. Find the UMP
test of
case n=3 and
versus
and evaluate the constants for the
.
The likelihood ratio for 1 versus
is large or equivalently if
rejects if
is large. To get level
that
and
take
where
make t
1.
page 213, number 7
To compute
independent. Now
you can compute
because X and Y are
and similarly for Y.
To compute the distribution of XY define U=XY and V=Y, say. Then you
should draw a picture to see what possible values U and V can have because
afte
Problems: Practice Problems
1.
Suppose
are iid
and
Assume the Xs are independent of the Ys.
(a)
Find complete and sufficient statistics.
are iid
The log likelihood is
It follows that
is complete and sufficient. There are many acceptable alternative one to
1.
Compute the characteristic function, cumulants and central moments for the
Poisson( ) distribution.
The cumulant generating function is
The cumulants are thus
for all r. The moments are unpleasant since you just have to recover the
moments from these s
STAT 801: Mathematical Statistics Probability Definitions Probability Space (or Sample Space): ordered triple (, F, P ). is a set (possible outcomes); elements are called elementary outcomes. F is a family of subsets (events) of with the property that F i
STAT 801: Mathematical Statistics Hypothesis Testing Hypothesis testing: a statistical problem where you must choose, on the basis of data X, between two alternatives. We formalize this as the problem of choosing between two hypotheses: H o : 0 or H1 : 1
1.
Suppose X and Y have joint density f(x,y). Prove from the definition of density
that the density of X is
.
I defined g to be the density of X provided
but
Notice that I have used the fact that f is the density of (X,Y) in the middle of
this.
2.
Suppose
Midterm 1: Solutions
Richard Lockhart October 18, 1996
1.
Suppose that X and Y are independent and that each has density, f, given
by
for t>0 and
1.
for t<0.
Find the joint density of
and V=X+Y. [4 marks]
Solving for X and Y we get
The Jacobian is then
2.
1.
Suppose that X and Y are independent and that each has density, f, given
by
for t>0 and f(t)=0 for t<0.
(a)
Find the joint density of U=X/Y and V=X+Y. [4 marks]
Solving for X and Y we get Y=V/(1+U)and X=UV/(1+U). The Jacobian is
then V/(1+U2) and the j
Solutions: Assignment 3
1.
Number 37 on page 216.
1.
For
the cdf of Y is clearly 0. For
the cdf is evidently 1.
For
(The cdf of X is trivial to find.) Differentiating we get that the density
of Y is
2.
The first two lines above are the same but now
for
so
1.
Suppose
Let
1.
are iid real random variables with density f .
be the X 's arranged in increasing order.
Find the joint density of
.
Let g be the joint density and
.
Since
we may take g to be 0 on
For
Divide by
in A choose so small that
. Then
and let t
1.
Page 398. Q 1.
One pivot is
which is Uniform on [0,1]. So
which leads to the interval
The event
can be rewritten as
whose probability is
Take an interval of the form aY,bY whose coverage probability is
and minimize b-a subject to the coverage being
sol
1.
Page 365. Q 21.
The mean of a Uniform
distribution is
find that
1.
is unbiased so is
. Since
equal to its variance which is
is
2.
make
all
.
provided that all
's are less than . To
's less than is
is
The bias of
. The mean
(and of
. This can be
and
is
1.
Compute the characteristic function, cumulants and central moments for the
Poisson( ) distribution.
The cumulant generating function is
The cumulants are thus
for all r. The moments are unpleasant since you just have to recover the
moments from these s