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
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
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 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
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
M372K SPRING 2014 UNIQUE NUMBER 57005 HOMEWORK 2
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5.6. Rayleigh Quotient 189
5.5A.3. Consider the eigenvectors of the matrix
6 4
A _ [ 1 3 ] .
(a) Show that the eigenvectors are not orthogonal.
(b) If the dot product of two vectors is dened as follows,
0 b = 016 + 02b2,
show that the eigenvectors are ort
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