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
11101
11000
10110
10011
01100
01011
00111
00000
Majori
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 = fo; i; e; a; ug
_ N = f1; 2; 3; 4; 5; :g (natural num
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 the following functions denes an Exponential Family dist
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 discrete random variable with pmf fX specied by
fX (x) =
k
x
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
Icfw_1,2,3,. (x)
1
expcfw_x log = h(x)c() expcfw_w()t(x)
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 = indicates equality in distribution (that
1
is, the left
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 sum telescopes. Noting that we must have FX (x) 1 as x
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, use the key stochastic convergence concepts for a sequen
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 continuous random variables, with joint pdf specied as
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 the blood test results for 471 male volunteers who were
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 = (y1 , . . . , yn )T with canonical parameters i is
l
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 and small, then, approximately
X
P oissonFG
with
m.
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 the rst derivative of the log-likelihood with
respect to
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 ranging from 53F to 81F. We seek to
assess the probabi
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 on the asymptotic theory of likelihood ratio tests: we
h
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 is dened by
A p tq
t
8
ds
V 1cfw_3 psq
The 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 this two factor model.
The R code for tting these model
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!
2
F1 eG
so that
VarfY HY I
-fY HY FY
2
F 1 e G
Also
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 at Baystate Medical Center, Springeld, Mass during
198
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 cancer registry records.
Tumor type: type
1
2
3
4
Hutchin
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 the degree to which it was processed when it was initial
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
statistics packages.
1. The following data relate to a 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 assuming a randomized block design (RBD) as in lectures
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
statistics packages.
1. The following data relate to a 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
objective of diagnosing whether any other measured variables
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 number of experimental units in the ith treatment group