lesson
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lesson
topics
total
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Part 1: Basics of probability theory and probability
models
1
30/9/2013
Foundations of
probability. Basic concepts.
2
2
2/10/2013
Basic combinatorics.
Stirling's Approximation. 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.
4
3
3/10/2013
Independent events.
Conditional probabilities in stochastic modeling:
gambler's ruin and probabilistic diffusion
models.
5
4
9/10/2013
Bertrand's Paradox.
First Borel-Cantelli lemma. Second Borel-Cantelli
lemma.
The transition from sample space to random variables.
Review of basic concepts and definitions on discrete
and continuous random variables.
7
5
10/10/2013
Uniform distribution. Buffon's needle.
Transformations of random variables.
8
6
14/10/2013
Mathematical
expectation. Dispersion (variance). Properties of
expectation and variance.
Chebyshev's inequality. From Chebyshev's inequality to
the weak law of large numbers. Other inequalities in
probability theory: Markov's inequality, generalized
Chebyshev's inequality.
10
7
16/10/2013
Strong law of large
numbers. Chernoff bound.
Probability models 1: uniform distribution, Bernoulli
distribution and binomial distribution 12
8
21/10/2013
Probability models 2:
Multinomial distribution. The Poisson distribution.
Examples. Cell plating statistics in biology.
14
9
23/10/2013
Probability models 3:
The exponential distribution. Paralyzable and
non-paralyzable detectors (application of the
exponential distribution). The De Moivre-Laplace
theorem and the Gaussian distribution. Properties of
the Gaussian distribution. The Error function.
Gaussian approximation of the Poisson
distribution.Transformations of random variables. Sum
of random variables (convolution).
16
10
30/10/2013
Probability models 4:
PDF of the 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. The multivariate normal
distribution.
18
11
31/10/2013
Lognormal distribution.
Gamma distribution. Cauchy distribution. Landau
distribution. Rayleigh distribution. Other
distributions.
19
12
6/11/2013
Jaynes' solution of
Bertrand's paradox.
Introduction to generating functions: examples.
21
13
7/11/2013
Probability generating
functions (PGF). PGF of the Poisson distribution. PGF
of uniform and binomial distributions. Poisson
distribution as limiting case of a binomial
distribution.
22
14
8/11/2013
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. 24
15
11/11/2013
The Central Limit Theorem (CLT). The
Berry-Esseen theorem.
26
16
13/11/2013
Additive and multiplicative processes.
Power-laws from the lognormal distribution.
Introduction to discrete-time stochastic processes.
Markov chains. Transient and persistent states in
Markov chains. 28
17
14/11/2013
Return probabilities. Transient and
persistent states in the 1D random walk.
29
18
18/11/2013
Transient and persistent states in
higher-dimensional random walks.Invariant
distribution. Time reversal and detailed balance.
30
Part 2: Introduction
to statistical inference
18
18/11/2013
Brief discussion of descriptive
statistics.
31
19
20/11/2013
Sample mean, sample
variance, estimate of covariance and correlation
coefficient. Statistics of sample mean for
exponentially distributed data.Limitations of the
standard deviation as a descriptor of the width of a
distribution. Shannon entropy and information
concentration in pdf's.
32
20
21/11/2012
Example of synthesis of
nonparametric estimators in the "German Tank Problem".
The Allan variance for noise processes with infrared
divergences.
33
21
27/11/2013
Introduction to the
Monte Carlo method. Pseudorandom numbers. Uniformly
distributed pseudorandom numbers. Transformation
method. Acceptance-rejection method.
35
22
28/11/2013
Examples: generation of
angles in the e+e- -> mu+mu- scattering; generation
of angles in the Bhabha scattering.
The structure of a complete MC program to simulate
low-energy electron scattering. 36
23
2/12/2013
Transformation method
and the trasformation of differential cross
sections.Early history of the Monte Carlo method.
Statistical bootstrap 38
24
4/12/2013
Maximum likelihood
method 1. Point estimators. Connection with Bayes'
theorem. Properties of estimators. Maximum likelihood
method.
40
25
5/12/2013
Maximum likelihood
method 2. Consistency of the maximum likelihood
estimators. Asymptotic optimality of ML estimators.
Bartlett's Identities.
41
26
9/12/2013
Maximum likelihood
method 3. Cramer-Rao-Fisher bound. Variance of ML
estimators. Information measures.
43
27
10/12/2013
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.
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.
Brief introduction to the Feldman-Cousins construction
(link to the FC
paper).
45
28
11/12/2013
Maximum likelihood
method 4. 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.
Hypothesis test, significance level. Examples.
Critical region. Construction of test statistics.
Neyman-Pearson lemma.
47
29
12/12/2013
Chi-square test.
Significance of a signal. Detailed analysis of the
statistical significance of a peak in spectral
estimation. Brief introduction to the test statistics
used by the LHC experiments.
48