Circuit Complexity
3.1 Boolean Logic
_ Objects: statements p, 1
_ Operators: _;^;:, etc
3.2 Logic Gates
Will insert logic gate diagrams later when I _gure how to insert images.
p q r p _ q (p _ q) _ r
1 Sets
1.1 De_nition
A set is a collection of distinct objects which are called the elements of the set.
Examples: We use a capital letter for sets.
_ A = fAlice;Bob;Claire;Eveg
_ B = fa; e; i; o; ug
MATH 556 - ASSIGNMENT 3
To be handed in not later than 5pm, 15th November 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1
(a) State whether each of t
MATH 556 - ASSIGNMENT 1
To be handed in not later than 5pm, 20th September 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose that X is a discr
MATH 556 - A SSIGNMENT 3
S OLUTIONS
1
(a) (i) This is not an Exponential Family distribution; the support is parameter dependent.
1 M ARK
(ii) This is an EF distribution with k = 1:
f (x|) =
where
Icf
MATH 556 - A SSIGNMENT 4
S OLUTIONS
1 By properties of the Gamma distribution, we can write
Vn =
(n 1)s2
n
2 Gamma
n1
2
n1 1
,
2
2
d
n1
= Vn =
Xi
i=1
d
where Xi 2 Gamma (1/2, 1/2), and the symbol = i
MATH 556 - A SSIGNMENT 1
S OLUTIONS
1. We have, for x = 1, 2, . . .
x
FX (x) =
x
t=1
but
k
t(t + 1)
fX (t) =
t=1
1
1
1
=
t(t + 1)
t
t+1
so, in fact
x
FX (x) = k
t=1
1
1
k
kx
=k
=
t
t+1
x+1
x+1
as the
MATH 556 - ASSIGNMENT 4
To be handed in not later than 5pm, 29th November 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
In the following questions, u
MATH 556 - ASSIGNMENT 2
To be handed in not later than 5pm, 11th October 2007.
Please hand in during lectures, to Burnside 1235, or to the Mathematics Ofce Burnside 1005
1. Suppose that X1 and X2 are
MATH 523 - ASSIGNMENT 2
To be handed in not later than 5pm, Friday 4th March 2011.
Please hand in during lectures, to Burnside 1225, or to the Mathematics Ofce Burnside 1005
The following data reect t
MATH 523: I TERATIVELY R EWEIGHTED L EAST S QUARES
In the exponential-dispersion family, with weights w1 , . . . , wn and identity dispersion function a() =
, the log-likelihood for independent data y
MATH 523: E XAMPLE - P OISSON DATA WITH O FFSET
A Poisson GLM can also be used in situations where a binomial model might seem more appropriate.
It is well known that if X BinomialFm, G, with m large
MATH 523: S OME KEY ASYMPTOTIC RESULTS
In a probability model with independent data y1 , . . . , yn , maximum likelihood estimation proceeds
by solving the likelihood equations, obtained by setting th
MATH 523: E XAMPLE - B INOMIAL DATA
The following data are counts of the number of failed O-rings (out of six) in twenty three launches
of the space shuttle Challenger at different ground temperatures
MATH 523: D EVIANCE COMPARISON WITH ESTIMATED DISPERSION
When is known, the Analysis of Deviance for nested models M0 M1 containing p0 p1 parameters
respectively, with deviances D0 and D1 , is based o
MATH 523: R ESIDUALS
Three types of residual are used for checking the t of a GLM
Pearson residual
y
p
ai i
p
V p i q
rP i
Anscombe residual
p
Apyiq a Apiq
Api q V pi q
p
p
rAi
'
where function A
MATH 523 - A SSIGNMENT 2: S OLUTIONS
(a) Five basic models can be tted:
M0
M1
M2
M3
M4
Null
age
test
age+test
age+test+age.test
Here model M4 is the saturated model, the most complicated we can t for
MATH 523 - A SSIGNMENT 1: S OLUTIONS
1 By direct calculation
-fY HY I
8
y
y 1
e y
F1 eGy!
F1 e eG
8
y1
Fy 1G!
F1 eG
y 1
Similarly
8
e y
-fY HY FY 1GI
y Fy 1 G
F1 eGy!
y 1
8
2 e y2
F1 eG y2 Fy 2G
MATH 523 - EXERCISES 1
These exercises are not for assessment
1 The birthwt data set included in the MASS library in the package R contains information on the
low birthweight status of 189 babies born
MATH 523: E XAMPLE - P OISSON DATA
An investigation of the incidence of the occurrences of four types of tumor at three different body
locations involved recording counts from 400 randomly sampled can
MATH 204 - ASSIGNMENT 1
Please Hand in Assignment in the Lecture on Friday 26th January.
A standard model of memory is that the degree to which the subject remembers verbal material is a
function of t
MATH 204 - ASSIGNMENT 3
Please Hand in Assignment in the Lecture on Friday 30th March.
For this assignment, all calculations can be done by hand with a calculator. However, you may use SPSS or other
s
MATH 204 - EXERCISES 2
These exercises are not for assessment
1 Download the SOIL.SAV dataset from the course website at
www.math.mcgill.ca/dstephens/204/Data/Soil.sav
(a) Repeat the analysis in SPSS
MATH 204 - ASSIGNMENT 3
Please Hand in Assignment in the Lecture on Friday 30th March.
For this assignment, all calculations can be done by hand with a calculator. However, you may use SPSS or other
s
MATH 204 - ASSIGNMENT 2
Please Hand in Assignment in the Lecture on Wednesday 7th March.
The study of lung expiratory pressure capacity in sufferers from cystic brosis is to be studied, with the
objec
MATH 204 - EXERCISES 1
These exercises are not for assessment
The following questions relate to a completely randomized design (CRD) with k treatment groups.
We use the following notation:
ni is the