lesson
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lesson
topics
total
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Part 1: Basics of probability theory and probability
models
1
14/10/2014
Introduction to the
course. Review of the basic concepts of
probability. Basic combinatorics. Stirling's
Approximation.
2
2
16/10/2014
Set theory
representation of probability space. Addition law for
probabilities of incompatible events. Addition law for
probabilities of non-mutually exclusive events.
Conditional probabilities. Bayes'
theorem. Independent events. Conditional
probabilities in stochastic modeling: gambler's ruin
and probabilistic diffusion models (1).
4
3
20/10/2014
Conditional
probabilities in stochastic modeling: gambler's ruin
and probabilistic diffusion models (2). Bertrand's
Paradox (1).
First Borel-Cantelli lemma.
6
4
23/10/2014
Second Borel-Cantelli
lemma.
The transition from sample space to random variables.
Review of basic concepts and definitions on discrete
and continuous random variables. Uniform
distribution. Buffon's needle. Transformations of
random variables. Brief review of mathematical
expectation, dispersion (variance), and of the
properties of expectation and variance. Chebyshev's
inequality. 8
5
27/10/2014
From Chebyshev's inequality to the
weak law of large numbers. Other inequalities in
probability theory: Markov's inequality, generalized
Chebyshev's inequality. Strong law of large numbers.
Chernoff bound.
10
6
29/10/2014
Very brief review of
common probability models. Examples: 1. The uniform
distribution; 2. the Bernoulli distribution; 3 the
binomial distribution; 4. the multinomial
distribution; 5. Poisson distribution; memoryless
random processes; Examples: lottery tickets and the
Poisson statistics; cell plating statistics in
biology.
12
7
5/11/2014
Distributions (ctd):
5. exponential distribution. Example: paralyzable and
non-paralyzable detectors; 6. the De Moivre-Laplace
theorem and the Gaussian distribution; 7. The
multivariate normal distribution.
14
8
6/11/2014
(recupero, aula B)
Transformations of
random variables. Sum of random variables
(convolution). Product of two random variables
(Mellin's convolution). Functions of random variables.
Approximate transformation of random variables: error
propagation. Linear (orthogonal) transformation of
random variables.
Other important distributions. 16
9
7/11/2014
(recupero, aula B)
Jaynes' solution of
Bertrand's paradox. The nature of randomness.
Randomness in a coin toss. Bell's inequalities.
18
10
10/11/2014
Introduction to
generating functions: examples. Probability generating
functions (PGF). PGF of the Poisson distribution. PGF
of uniform and binomial distributions. Poisson
distribution as limiting case of a binomial
distribution.
20
11
12/11/2014
PGF of the
Galton-Watson branching process. Photomultiplier
noise.
Characteristic functions. Moments of a distribution.
Skewness and kurtosis. Mode and median. Properties of
characteristic functions.
The Central Limit Theorem (CLT). 22
12
19/11/2014
The Berry-Esseen
theorem. Additive and multiplicative processes. Stable
distributions.
Introduction to discrete-time stochastic processes.
Markov chains. Transient and persistent states in
Markov chains. 24
13
20/11/2014
(recupero, aula B) Transient and
persistent states in higher-dimensional random walks.
Invariant distribution (1).
26
14
24/11/2014
Invariant distribution (2). Time
reversal and detailed balance. Transient and
persistent states in higher-dimensional random
walks.Invariant distribution. Time reversal and
detailed balance. Continuous time Markov Processes.
Boltzmann's H-theorem. Hidden Markov Models.
28
Part 2: Introduction
to statistical inference
15
26/11/2014
Descriptive statistics.
Sample mean, sample variance, estimate of covariance
and correlation coefficient. Statistics of sample mean
for exponentially distributed data.
Example of synthesis of nonparametric estimators in
the "German Tank Problem".
30
16
1/12/2014
Order statistics.
Limitations of the standard deviation as a descriptor
of the width of a distribution.
The Allan variance for noise processes with infrared
divergences. Shannon entropy and information
concentration in pdf's.
Introduction to the Monte Carlo method.
32
17
3/12/2014
Pseudorandom numbers.
Uniformly distributed pseudorandom numbers.
Transformation method. Acceptance-rejection method.
Examples: generation of angles in the e+e- ->
mu+mu- scattering; generation of angles in the Bhabha
scattering.
34
18
10/12/2014
The structure of a
complete MC program to simulate low-energy electron
scattering.
Transformation method and the trasformation of
differential cross sections.Early history of the Monte
Carlo method.
Statistical bootstrap. 36
19
11/12/2014
(recupero, aula B)
Maximum likelihood
method 1. Point estimators. Connection with Bayes'
theorem.
37
20
16/12/2014
(recupero, aula B, ed. A)
Maximum likelihood
method 2. Properties of estimators. Consistency of the
maximum likelihood estimators. Asymptotic optimality
of ML estimators.
39
21
17/12/2014
Maximum likelihood
method 3. Bartlett's Identities. Cramer-Rao-Fisher
bound. Variance of ML estimators. Introduction to
Shannon's entropy.
41
22
7/1/2015
Maximum likelihood
method 4. Information measures based on the Shannon's
entropy: Kullback-Leibler divergence, Jaynes distance,
Fisher information. Introduction to confidence
intervals.
43
23
8/1/2015 (recupero)
Maximum likelihood
method 5. 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.
Graphical method for the variance of ML estimators.
45
24
9/1/2015 (recupero)
Maximum likelihood
method 6. Extended maximum likelihood. Examples.
Introduction to ML with binned data. Example with two
channels. Other examples of ML with binned data: decay
rate in radioactive decay; exponent of power-law.
Very brief overview of chi-square and least squares
fits, chi-square distribution, weighted straight line
fits, general least squares fits, least squares
fitting of binned data, and nonlinear least squares.
Fit quality and dimension of parameter space.
47
25
12/1/2015
Maximum likelihood
method 7. Hypothesis test, significance level.
Examples. Critical region. Construction of test
statistics. Neyman-Pearson lemma. Chi-square test.Significance
of a signal. Detailed analysis of the statistical
significance of a peak in spectral estimation. The
Feldman-Cousins construction (link
to the FC paper). Brief introduction to the
test statistics used by the LHC experiments.
49