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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 the moment, as unexplained, primitive notions
of the theo
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 kind of universal random variable
in probabilistic mode
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
this by a normally distributed random variableThis
is the
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 we will systematically denote by capital letters, e.g.
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-style
dierentials, postponing the mathematically rigorou
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 = y)fY (y) dy.
c
Again, the concept of conditional proba
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 another one. Since a Brownian motion (Wt )t0 (or any s
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 ])/N t s as N ,
so that, for big N , the CLT implies that
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 nancial asset
prices assumes that the log-returns of a gi
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
distribution, with the same mean and variance as X.
F Exam
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
hormones to adminster. What should the experimenter do
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 electrons. (10 pts.)
OH
O
H+/H2O
OH
6. The reaction 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 abbreviations or
acronyms for the reagents-write them
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 will display lordosis and attempt to
mount any rat it sees
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 arteries, causing vascular inflammation and
atheroscleros
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 male/female sexual
behaviour
b) The rat will display lord
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. records an action potential through a single channel.
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, micro-economic and business-specific factors, e.g., flexibi
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
operative definition, in a computing context, might be to
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
between Brownian motion and this heat equation.
The rel
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, one is smaller than the
other. Which is a possible explan
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 this have on a) the size of the postsynaptic EPSP when 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 blue opscan sheet included with this exam. Total point
Hormonal Causes of Sexual Behaviour
1.
2.
3.
4.
83
Hormonal Causes of Sexual Behaviour
Summary
For rats to exhibit female sexual behaviour you need the
appropriate female sexual hormones (estradiol and
progesterone) as an adult
For rats to exhibit male se
Mathematical Methods I, Autumn 2009
and
var(X) =
n
X
15
(aj X )2 pj .
j=1
What about (2.13) when g has a jump in x = a1 and is right-continuous there?
And what if it is left-continuous? (Take a1 = 0, to simplify.)
Exercise 2.14. Compute the mean and varia
Mathematical Methods I, Autumn 2009
37
Exercise 3.10. Let (Wt )t0 be a Brownian motion and let , be two real
numbers, with > 0. The new process (Xt )t0 dened by Xt = t + Wt is
called Brownian motion with drift and volatility .
(a) Determine the distributi
Mathematical Methods I, Autumn 2009
17
Here V 1 x stands for the inverse of V applied to x = (x1 , . . . xN ) RN , and
h, i stands for the Euclidean inner product on RN :
hx, yi = x1 y1 + + xN yN
(often written as xt y, where t stands for transpose).
Rema
Mathematical Methods I, Autumn 2009
19
F Remark 2.25. Having the right-hand side of (2.23) for a sufficiently large
class of functions g and h, for example, for all bounded continuous functions, is
in fact equivalent to Definition 2.23.
If we recall that