Quiz 7
1. Write the Pareto model; i.e.
f (x; ) =
(1 + x)(1+)
for x > 0 and > 0, in terms of the exponential family, saying what c(x) is,
t(x) and b(). Hence, what is the sucient statistic for based on a sample
X1 , . . . , X n .
2. Do the same as in quest
Quiz 3
Suppse X1 , . . . , Xn are independent normal random variables with unknown
2
mean 0 and known variance 0 .
For a set of (wi ), with each 0 < wi < 1 and
n
wi = 1,
i=1
it is proposed to use the estimator
n
wi X i
=
i=1
for 0 .
Note that the usual es
Lecture 1 Summary
The rst lecture will be introducing the ideas behind mathematical statistics.
There is a mechanism which we want to learn about and the mechanism
provides us with obervations. We will write these observations as
X1 , . . . , X n
which as
Quiz 2
Part 1. Suppose
X1 , . . . , X n
are random outcomes from a normal distribution with known mean 0 and
2
unknown variance 0 .
The density function for such a normal model with general variance term
2
is given by
1 2 2
1
e 2 x / .
f (x; 2 ) =
2 2
Quiz 2 Answer
Part 2. Suppose (X1 , . . . , Xn ) come from a Poisson distribution with parameter 0 > 0; i.e.
f (x; 0 ) =
x
0 0
e ,
x!
for x = 0, 1, 2, . . . .
Write down the likelihood function for ; i.e. L(), maximize this to obtain
a proposal estimator
Quiz 1
Suppose
X1 , . . . , X n
are random outcomes from a normal distribution with unknown mean 0 and
2
known variance 0 . This means that for each Xi , the expected value of Xi
2
is 0 ; i.e. E(Xi ) = 0 , and the variance of each Xi is Var(Xi ) = 0 .
We
Lecture 12 Summary
Following on from Lecture 11, the key test of interest is in testing the
hypothesis
H0 : 0 = 0 vs H1 : 0 = 0.
This is the most important test since it tells us whether the xi has any eect
on the outcome yi at all. For if 0 = 0 then ther
Lecture 3 Summary
2
First, we will look at the point estimation of 0 ; looking at the maximum
likelihood value and so answering Part 1 of Quiz 2.
Next, we will nd an interval estimate of 0 , to get some idea of how
accurate = X is. The variance indicates
Lecture 10 Summary
We have the class of density given by
f (x, ) = c(x) expcfw_x b().
For the normal model we have b() = 1 2 , for the Bernoulli model we have
2
b() = log(1 + e ), and for the Poisson model we have b() = e . Obviously
there are many choice
Quiz 5 Answer
In the normal example with 0 unknown and 0 known, for the test,
H0 : 0 = 0 vs H1 : 0 = 0,
H0 is rejected at the level of signicance if
|T | =
X
> z/2 .
0 / n
The Power function at is the probability of rejecting H0 when is the true
value fo
Quiz 3 Answer
Suppse X1 , . . . , Xn are independent normal random variables with unknown
2
mean 0 and known variance 0 .
For a set of (wi ), with each 0 < wi < 1 and
n
wi = 1,
i=1
it is proposed to use the estimator
n
wi X i
=
i=1
for 0 .
Note that the u
Quiz 4 Answer
Suppse X1 , . . . , Xn are independent normal random variables with unknown
2
mean 0 and known variance 0 .
1. What is the smallest sample size n needed in order for the 100(1 )%
condence interval for 0 to be of size L0 , for some L > 0.
2.
Quiz 8
For the model
yi N ( xi , 2 )
with 0 known (so this is the model we did in lectures but with 0 = 0
known), write down the likelihood function for and hence nd .
Find E() and Var() and hence nd a 100(1 )% condence interval
for 0 .
Thus, what is the
Quiz 6
2
1. Suppose, X1 , . . . , Xn are i.i.d. from N (1 , 1 ) and Y1 , . . . , Ym are i.i.d.
2
from N (2 , 2 ). Also the two sequences are independent of each other. What
is the distribution of
X Y,
where X = n1 (X1 + + Xn ) and Y = m1 (Y1 + + Ym ).
H
Quiz 8 answer
For the model
yi N ( xi , 2 )
with 0 known (so this is the model we did in lectures but with 0 = 0
known), write down the likelihood function for and hence nd .
Find E() and Var() and hence nd a 100(1 )% condence interval
for 0 .
Thus, what
Quiz 7 Answer
The Pareto density with the old parameter is written as
f (x, ) =
(1 + x)(1+)
for x > 0 and > 0. This can be written as
f (x, ) =
1
expcfw_ log(1 + x) + log ,
1+x
so t(x) = log(1 + x), c(x) = 1/(1 + x) and b() = log .
The gamma density can b
Quiz 9
Consider again, as in Quiz 8, the model
2
yi N ( xi , 0 )
with 0 known.
Dene
yi = xi ,
where is given in the answer to Quiz 8, and
ri = y i y i .
Show that for all i it is that
E(ri ) = 0
and nd
Var(ri ).
Hence what is the distribution of
ri =
ri
.
Quiz 5
In the normal example with 0 unknown and 0 known, for the test,
H0 : 0 = 0 vs H1 : 0 = 0,
H0 is rejected at the level of signicance if
|T | =
X
> z/2 .
0 / n
The Power function at is the probability of rejecting H0 when is the true
value for 0 . F
Quiz 6 Answer
Question 1 we did in class.
2. Back to the normal model with a single sample X1 , . . . , Xn which are i.i.d.
2
2
from N (0 , 0 ), we have interest in 0 and the key result is
2 =
n1
(n 1)S 2
.
2
0
2
A one sided condence interval for 0 is of
Quiz 4
Suppse X1 , . . . , Xn are independent normal random variables with unknown
2
mean 0 and known variance 0 .
1. What is the smallest sample size n needed in order for the 100(1 )%
condence interval for 0 to be of size L0 , for some L > 0.
2. It is s
Lecture 11 Summary
Suppose we have n individuals and each individual has a piece of information associated with them, e.g. height, and is recorded as xi . This is
xed. Now normal random outcomes yi from each individual are recorded
and each individual out
Lecture 9 Summary
Now we move away from the normal distribution, and the general class
of density model we now consider is known as the exponential family. We
start with the Bernoulli model.
In this model, if X is a random outcome from a Bernoulli model w
Lecture 8 Summary
Continuing on from the previous lecture, we want now to consider X1 , . . . , Xn
as coming from a normal distribution with unknown mean 1 and unknown
2
variance 1 , and Y1 , . . . , Ym coming from a normal distribtuion with mean 2
2
and
M372K SPRING 2014 UNIQUE NUMBER 57005 HOMEWORK 2
This homework is due on Thursday, February 6, 2014, before lecture begins. Late
homework is not accepted. In order to receive credit for this assignment you must write your
name and UT eid at the top of fir
M372K SPRING 2014 UNIQUE NUMBER 57005 HOMEWORK 4
This homework is due on Thursday, February 27, 2014, before lecture begins. Late
homework is not accepted. In order to receive credit for this assignment you must write your
name and UT eid at the top of rs
M372K SPRING 2014 UNIQUE NUMBER 57005 HOMEWORK 5
This homework is due on Thursday, March 6, 2014, before lecture begins. Late homework is not accepted. In order to receive credit for this assignment you must write your name and
UT eid at the top of rst pa
M372K SPRING 2014 UNIQUE NUMBER 57005 HOMEWORK 10
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homework is not accepted. In order to receive credit for this assignment you must write your
name and UT eid at the top of first
(15 pts total)
1.4.1 (h)
The equilibrium solution is independent of time ) u(x, t) = u(x)
(1)
) ut (x, t) 0
0 = 1 uxx (x) + Q = uxx + 0 = uxx
(4)
Integrate w.r.t x to get: ux (x) = c
By the initial condition that ux (L) = , we have ux (x) = = c.
(3)
Integ