lesson
date
lesson
topics
total
time
(%
of
total)
Part 1: Basics of probability theory and probability models
1
18/10/2010
Foundations
of
probability.
Basic
concepts.
The
Stirling's
Approximation.
Set
theory
representation
of probability space. Addition law for probabilities of incompatible
events.
2
(4
%)
2
20/10/2010
Addition
law
for
probabilities
of
non-mutually
exclusive
events.
Assignement
problems. First Borel-Cantelli lemma. Conditional probabilities. Bayes
theorem.
4
(8
%)
3
21/10/2010
Dependent
event,
Bayes
theorem.
Second
Borel-Cantelli
lemma.
Discrete
and
continuous
random
variables. Uniform distribution. Buffon's needle.
6
(13
%)
4
27/10/2010
Discrete
and
continuous
random
variables.
Transformations
of
random
variables.
Sum
of random variables
(convolution), product of random variables (Mellin's convolution).
Mathematical expectation. Dispersion (variance). Properties of
expectation and variance. Chebyshev's inequality.
8
(17
%)
5
4/11/2010
Using
Chebyshev's
inequality
to
prove
the
weak
law of large numbers.
Experimental planning using bounds derived from Chebyshev's inequality.
Other inequalities in probability theory: Markov's inequality, Chernoff
bound. Approximate transformation of random variables: error
propagation.
10
(21
%)
6
6/11/2010
(saturday)
Error
propagation.
Linear
(orthogonal)
transformation
of
random
variables.
Some
important
probability
models: uniform distribution, binomial distribution and multinomial
distribution.
12
(25
%)
7
8/11/2010
Probability
models
2:
the
multinomial
distribution,
the
Poisson distribution, the
exponential distribution.
14
(29
%)
8
10/11/2010
Probability
models
3:
Paralyzable
and
non-paralyzable
detectors
(application of the
exponential distribution). The
De Moivre-Laplace theorem and the Gaussian distribution.
16
(33
%)
9
11/11/2010
Lognormal
distribution.
Gamma
distribution.
Cauchy
distribution.
Landau
distribution.
Introduction to generating function: examples.
18
(38%)
10
15/11/2010
Generating
functions.
Probability
generating
functions
(PGF).
PGF of uniform,
binomial and Poisson distributions. Poisson distribution as limiting
case of a binomial distribution. PGF of the Galton-Watson branching
process (application to photomultipliers).
20
(42
%)
11
17/11/2010
Photomultiplier
noise.
Characteristic
functions.
Moments
of
a distribution. Skewness
and kurtosis. Mode and median.
22
(46
%)
12
18/11/2010
More
on
characteristic
functions.
The
Central
Limit Theorem (CLT). Additive
and multiplicative processes. Power-laws from the lognormal
distribution.
24
(50
%)
13
24/11/2010
Cumulants.
Introduction
to
discrete-time
stochastic
processes.
Markov
chains.
26
(54
%)
14
25/11/2010
Transient
and
persistent
states
in Markov chains. Transient and persistent states
in the 1D random walk. Invariant distribution. Time reversal and
detailed balance.
28
(58
%)
Part
2:
Introduction
to
statistical
inference
15
29/11/2010
The
Monte
Carlo
method.
Pseudorandom
numbers.
Uniformly
distributed
pseudorandom
numbers.
30
(63
%)
16
2/12/2010
Transformation
method.
Acceptance-rejection
method.
Examples: generation of angles in the e+e- -> mu+mu-
scattering; generation of angles in the Bhabha scattering.
32
(67
%)
17
6/12/2010
Statistical
bootstrap.
34
(71
%)
18
9/12/2010
Descriptive
statistics.
Sample mean, sample variance, estimate of
covariance and correlation coefficient. Statistics of sample mean for
exponentially distributed data. Confidence intervals and confidence
level. Confidence intervals for the sample mean of exponentially
distributed samples. Confidence intervals for the correlation
coefficient of a bivariate Gaussian distribution from MC simulation.
36
(75
%)
19
13/12/2010
Maximum
likelihood
method
1.
Point
estimators. Connection with Bayes' theorem. Variance of ML
estimators. The Cramˇr-Rao-Fisher bound.
38
(79
%)
20
15/12/2010
Maximum
likelihood
method
2.
Asymptotic
optimality of ML estimators. Graphical method for the
variance of ML estimators. Example with two channels. Introduction to
ML with binned data.
40
(83
%)
21
20/12/2010
Extended
ML. ML with binned data. Chi-square and least
squares
fits.
Chi-square distribution. Weighted straight line fits.
43
(90
%)
23
22/12/2010
General
least squares fits. Hypothesis
test,
significance level. Examples. Critical
region. Construction of test statistics. Neyman-Pearson lemma.
45
(94
%)
24
23/12/2010
Chi-square
and multidimensional confidence intervals. Least squares fitting of
binned data. Chi-square test. Significance of a signal. Detailed
analysis of the statistical significance of a peak in spectral
estimation.
Discussion of possible exam topics.
47
(98
%)