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
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 fi
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.
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
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 for
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
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
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
STAT 801: Mathematical Statistics
The Multivariate Normal Distribution
Def n: Z R1 N (0, 1) iff
2
1
fZ (z) = ez /2
2
Def n: Z Rp M V N (0, I) if and only if Z = (Z1 , . . . , Zp )t with the Zi indepen
STAT 801: Mathematical Statistics
Independence, conditional distributions
So far density of X specified explicitly. Often modelling leads to a specification in terms of marginal and
conditional distri
STAT 801: Mathematical Statistics
Course outline:
Distribution Theory.
Basic concepts of probability.
Distributions
Expectation and moments
Transforms (such as characteristic functions, moment ge
STAT 801: Mathematical Statistics
Normal samples: Distribution Theory
Theorem: Suppose X1 , . . . , Xn are independent N (, 2 ) random variables. Then
(sample mean)and s2 (sample variance) independen
STAT 450: Statistical Theory
Course Overview
Text coverage
Covering Ch 110 of Casella and Berger.
You are responsible for material in Ch 14
from STAT 330.
Please read the notes on my web page for
O
STAT 801: Mathematical Statistics
Statistics versus Probability
Standard view of scientific inference has a set
of theories which make predictions about the
outcomes of an experiment:
Theory
A
B
C
Pre
STAT 450: Statistical Theory
Distribution Theory
Reading in Casella and Berger: Ch 2 Sec 1,
Ch 4 Sec 1, Ch 4 Sec 6.
Example: Why does t-statistic have t distribution?
Ingredients: Sample X1, . . . ,
STAT 801: Mathematical Statistics
Distribution Theory
Basic Problem: Start with assumptions about f or CDF of random vector X = (X1 , . . . , Xp ). Define
Y = g(X1 , . . . , Xp ) to be some function o
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 th
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
Th
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 s
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
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 f
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
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
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 ma
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)
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
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
F
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 probabilit
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
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