-~f~_oqlctlat. (tt'~J. '-fI
.!:.!-
_
-t-'-'~ t. P.e-;~diC. ~ h l t.-_. I 6,tJ\t_Cllet''l\~_f- 'tc. e f es _ vet b _ _ _ _ _1 _ W_ID_~l.!2lJ.=IJ.-=.:b~-:- _ .m
_ _~1Ld Itl~ .
~fM-fuqj-=:.kl'1~vJe 'cfw_
Chapter 2
Review of probability
theory (19th-century style)
2.1
Real random variables
We shall initially take an informal approach to probability theory, using probability and random variables, for th
Mathematical Methods I, Autumn 2009
9
The condition that P( < X < ) follows from the classical integral
Z
2
ex /2 dx = 2;
see any reasonable calculus textbook.
Standard normal random variables are a
Mathematical Methods I, Autumn 2009
23
be regarded as the sum of a lot of small independent but basically identically
distributed random influences, the CLT suggests that it is reasonable to model
thi
Mathematical Methods I, Autumn 2009
5
A case in point will be stochastic calculus, which aims to extend the notion
of derivative to the case where x and y above are replaced by random variables (which
Chapter 4
A crash course in It
o
calculus
In this chapter we introduce the basic rules of stochastic calculus or It
o calculus.
We will do this using an intuitive approach which is based on calculus-s
Mathematical Methods I, Autumn 2009
21
We record the following useful formulas:
Z bZ d
fX,Y (x, y) dx dy
P(a < X < b and c < Y < d) =
c
a
Z
b
Z
d
P(X = x | Y = y)fY (y) dx dy
=
c
a
Z
=
d
P(a X b | Y =
Mathematical Methods I, Autumn 2009
3.6
35
Conditional expectations and a first encounter
with martingales
In section 2.3 we introduced conditional probabilities of one random variable
with respect to
Mathematical Methods I, Autumn 2009
27
More generally, if t > s, let us consider the dierence
(N )
Wt
[tN ]
Ws(N ) = N 1/2
Xj .
(3.7)
[sN ]+1
We can easily check that this has variance ([tN ] [sN ])/
Chapter 3
Brownian motion
3.1
Introducing Brownian motion
Brownian motion has become one of the fundamental building blocks of modern
quantitative nance. Indeed, the basic continuous-time model for na
Mathematical Methods I, Autumn 2009
13
If the excess kurtosis is positive, the pdf of X is interpreted to have more
probability mass in the tails, or fatter tails, than that of a comparable normal
dis
Answer: MCQ 6
A female rat is given injections of testosterone at birth, and as
an adult is ovariectomized. The experimenter wants her to
display normal lordosis behaviour, but cannot figure out which
Parquette
Chemistry 252
Second Examination
Winter 2012
Name_
5. Draw a detailed mechanism for the following transformation showing all intermediates and an
arrow pushing formalism to show the flow of
Parquette
Chemistry 252
Second Examination
Winter 2012
Name_
Br2
H2O (solvent)
N
Li
1.
H
2.
O
3. H2O
1. O3
2. Me2S
O
O
9.
O
H
Provide reagents to complete the following transformations. Do not use any
MCQ 5
What happens if a male rat is castrated as an
adult, and is then given E+P injections
throughout the rest of his life?
a) The rat will display neither male/female
sexual behaviour
b) The rat wil
LAST NAME_FIRST NAME_ Student #_
6) What is atherosclerosis and which air pollutant is most closely related to this disease?
(4 points)
A study indicates that PM2.5 leads to high plaque deposits in ar
MCQ 4
What happens if you castrate a male rat immediately
after birth, give it an injection of E+P shortly after, and
then administer testosterone to it as an adult?
a) The rat will display neither ma
46. Patch clamping
A. aids in remyelination of a neuron.
B. fixes a break in the neuron plasma membrane.
C. records ion movements through ion channels.
D. inhibits impulse conduction along an axon.
E.
Chapter 1
Introduction
Market prices of liquidly traded financial assets depend on a huge number of
factors: macro-economic ones such as interest rates, inflation, balanced or unbalanced budgets, micr
Mathematical Methods I, Autumn 2009
11
(Such infinitesimals do not really exist, but we think of them as numbers which
are so small that their squares may be safely neglected in any computation. An
op
Mathematical Methods I, Autumn 2009
29
ow of heat is described by a certain partial dierential equation called the heat
equation and, as we will see in chapter 6, there indeed exists a close relation
BioNB2220
Cornell University
4.
(6 pts) Two EPSPs are triggered at two different dendrites simultaneously on the same neuron.
They start out the same amplitude, but by the time they reach the soma, on
BioNB2220
Cornell University
Exam 1
Feb. 29, 2012
13.
(6 pts) You are studying a mutant mouse with twice as many voltage-sensitive calcium
channels in its nerve terminals as normal. What effect will t
BioNB2220 Introduction to Neuroscience
Cornell University
Exam 1
Feb 22, 2013
PART 1: MULTIPLE CHOICE
Pick the best answer from among those given. Blacken your answer on the corresponding place on
the