lesson
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lesson topics
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Part 1: Basics of probability theory and probability models
1
29/9/2015
Introduction to the course.
Review of the basic concepts of probability. Basic
combinatorics. Stirling's Approximation. Set theory
representation of probability space. Addition law for
probabilities of incompatible events.
2
2
1/10/2015
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
6/10/2015
Conditional probabilities in
stochastic modeling: gambler's ruin and probabilistic
diffusion models (2). Bertrand's Paradox (1).
First Borel-Cantelli lemma. Second Borel-Cantelli lemma. 6
4
8/10/2015
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. 8
5
13/10/2015
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. Strong law of large
numbers. Chernoff bound.
10
6
20/10/2015
Moment generating function.
Bernoulli (indicator) random variables. Proof of the Chernoff
bound with applications (ctd.).
12
7
22/10/2015
Brief review of common
probability models. 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.
14
8
27/10/2015
Distributions (ctd): 5.
exponential distribution. Example: paralyzable and
non-paralyzable detectors; 6. the De Moivre-Laplace theorem
and the Gaussian distribution;
16
9
29/10/2015
Distributions (ctd): 7. The
multivariate normal distribution. 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.
18
10-11
6/11/2015
(recupero, aula B, ore 9-11, 11-13)
Distributions (ctd): other
important distributions. Example: the distribution of
nearest-neighbor distance.
Jaynes' solution of Bertrand's paradox. The nature of
randomness. Randomness in a coin toss. Bell's inequalities and
quantum probabilities.
Introduction to generating functions: examples. 22
12
10/11/2015
Probability generating
functions (PGF). PGF of the Poisson distribution. PGF of
uniform and binomial distributions. Poisson distribution as
limiting case of a binomial distribution.
PGF of the Galton-Watson branching process. Photomultiplier
noise.
24
13
12/11/2015
Photomultiplier noise. (ctd.)
Characteristic functions. Moments of a distribution. Skewness
and kurtosis. Mode and median. Properties of characteristic
functions.
The Central Limit Theorem (CLT). The Berry-Esseen
theorem. Additive and multiplicative processes.
26
Part
2: Introduction to statistical inference
14
17/11/2015
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". Order statistics. 28
15
19/11/2015
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. Early history of the
Monte Carlo method. 30
16
24/11/2015
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.
32
17
26/11/2015
The structure of a complete MC
program to simulate low-energy electron scattering.
Transformation method and the trasformation of differential
cross sections.
Statistical bootstrap. 34
18
1/12/2015
Statistical bootstrap (ctd.).
Maximum likelihood method 1. Point estimators. Connection with
Bayes' theorem. 36
19
3/12/2015
Maximum likelihood method 2.
Properties of estimators. Consistency of the maximum
likelihood estimators. Asymptotic optimality of ML estimators.
Maximum likelihood method 3. Bartlett's Identities.
Cramer-Rao-Fisher bound. Variance of ML estimators.
Introduction to Shannon's entropy.
38
20-21
11/12/2015
(recupero, aula B, ore 9-11, 11-13)Maximum likelihood method 4.
Information measures based on the Shannon's entropy:
Kullback-Leibler divergence, Jeffreys distance, Fisher
information. Introduction to confidence intervals.
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. 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. 42
22
15/12/2015
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. Chi-square and
chi-square tests.
44
23
17/12/2015
Hypothesis tests, significance
level. Examples. Critical region and acceptance region. Errors
of the first and of the second kind. p-value and rejection of
the null hypothesis. Chi-square as a test statistic.
Neyman-Pearson lemma. Significance of a signal. Detailed
analysis of the statistical significance of a peak in spectral
estimation.
46
24
21/12/2015
(recupero, aula C 15-17)
Detailed analysis of the Neyman
construction of confidence intervals (link
to the Neyman paper). Confidence intervals and the
Feldman-Cousins construction (link
to the FC paper, link
to the presentation).
48
25
22/12/2015
"Statistical topics at the
LHC", talk given by Dr. D. Tonelli (link
to the presentation).
50