lesson
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Part 1: Probability theory and probability models
1
4/10/2016
Introduction to the course.
Review of the basic concepts of probability. Basic
combinatorics.
2
2
6/10/2016
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.
4
3
11/10/2016
Overview of random walk models
in physics. Conditional probabilities in stochastic modeling:
gambler's ruin and probabilistic diffusion models. Discrete
and continuous forms of gambler's ruin. Bertrand's Paradox
(1). Introduction to the Borel-Cantelli lemmas with proof of
some auxiliary theorems.
6
4
13/10/2016
First Borel-Cantelli lemma.
Second Borel-Cantelli lemma. Statistical independence and
dimensional reduction. 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 and its properties.8
5
18/10/2016
Review of dispersion (variance) and its
properties. 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.
Moment generating function. Bernoulli (indicator) random
variables. Proof of the Chernoff bound.10
6
25/10/2016
Applications of the Chernoff
bound: more specific forms of the Chernoff bound; application
to network routing.
11
7
27/10/2016
Brief review of common
probability models. 1. The uniform distribution; 2. the
Bernoulli distribution; 3 the binomial distribution; 4. the
geometric distribution; 5. the negative binomial distribution.
5. the hypergeometric distribution;
13
8
8/11/2016
Distributions (ctd): 6. the
multinomial distribution; 7. Poisson distribution; memoryless
random processes; Examples: lottery tickets and the Poisson
statistics; cell plating statistics in biology; 8.
exponential distribution. Example: paralyzable and
non-paralyzable detectors.
15
9
10/11/2016
Distributions (ctd): 9. short
overview of the De Moivre-Laplace theorem and the Gaussian
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.
17
10
15/11/2016
Linear (orthogonal)
transformation of random variables. The multivariate normal
distribution. Other important distributions (lognormal, gamma,
beta, Rayleigh, logistic, Laplace, Cauchy). Example: the
distribution of nearest-neighbor distance.
19
11
17/11/2016
Distribution of
nearest-neighbor distance (ctd.). Overview of Jaynes' solution
of Bertrand's paradox.
Introduction to generating functions.
21
12
22/11/2016
Generating functions.
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
24/11/2016
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.
25
Part
2: Statistical inference
13
24/11/2016
Descriptive and exploratory statistics. Sample mean, sample
variance, estimate of covariance and correlation coefficient.
PDF 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. Box plots. Outliers. Violin plots. Rug
plots. Kernel density plots.
27
14
29/11/2016
Introduction to the Monte Carlo
method. Early history of the Monte Carlo method. Pseudorandom
numbers. Uniformly distributed pseudorandom numbers.
Transformation method. Transformation method and the
trasformation of differential cross sections.
Acceptance-rejection method. Examples: generation of angles in
the e+e- -> mu+mu- scattering; generation of angles in the
Bhabha scattering.
30
15
1/12/2016
The structure of a complete MC
program to simulate low-energy electron transport.
Statistical bootstrap.
33
16
6/12/2016
Maximum likelihood method 1.
Point estimators. Connection with Bayes' theorem. Properties
of estimators. Consistency of the maximum likelihood
estimators. Asymptotic optimality of ML estimators. Bartlett's
Identities. Cramer-Rao-Fisher bound.
36
17
13/12/2016
Maximum likelihood method 2.
Variance of ML estimators. Introduction to Shannon's entropy.
Information measures based on the Shannon's entropy:
Kullback-Leibler divergence, Jeffreys distance, Fisher
information.
Introduction to confidence intervals. Confidence intervals and
confidence level. Detailed analysis of the Neyman construction
of confidence intervals (link
to the Neyman paper). Confidence intervals for the
sample mean of exponentially distributed samples. 39
18
15/12/2016
Confidence intervals for the
sample mean of exponentially distributed samples. Confidence
intervals for the correlation coefficient of a bivariate
Gaussian distribution from MC simulation. Maximum likelihood
method 3. More properties of the likelihood function.
Graphical method for the variance of ML estimators.
41
19
21/12/2016
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. 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.
43
20
22/12/2016
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. Confidence intervals and the
Feldman-Cousins construction (link
to the FC paper).
45
23
10/1/2017
Final seminar by Diego Tonelli:
Statistical Methods for the Large Hadron Collider (link
to slides).
47