lesson
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lesson
topics
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Part 1: Basics of probability theory and probability
models
1
2/10/2012
Foundations of
probability. Basic concepts.
2
2
3/10/2012
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. Dependent events, Bayes' theorem.
4
3
4/10/2012
Example: gambler's ruin
and probabilistic diffusion models.
5
4
9/10/2012
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
11/10/2012
Uniform distribution. Buffon's needle.
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.
8
6
16/10/2012
Chebyshev's
inequality. From Chebyshev's inequality to the weak
law of large numbers. Experimental planning using the
bounds derived from Chebyshev's inequality. Other
inequalities in probability theory: Markov's
inequality, generalized Chebyshev's inequality. Strong
law of large numbers. Chernoff bound.
10
7
17/10/2012
Probability models 1:
uniform distribution, binomial distribution and
multinomial distribution.
12
8
23/10/2012
Probability models
2: The Poisson distribution, the exponential
distribution. Paralyzable and non-paralyzable
detectors (application of the exponential
distribution). The De Moivre-Laplace theorem and the
Gaussian distribution.
13
9
24/10/2012
Probability models
3: Properties of the Gaussian distribution. The
Error function. Gaussian approximation of the Poisson
distribution. Transformations of random variables: pdf
of the sum of two random variables.
15
10
25/10/2012
Probability models
4: Transformations of random variables. PDF of
the product of two random variables. Functions of
random variables. Approximate transformation of random
variables: error propagation.
16
11
30/10/2012
Probability models
5: Linear (orthogonal) transformation of random
variables. The multivariate normal distribution.
Lognormal distribution. Gamma distribution. Cauchy
distribution. Landau distribution.
18
12
31/10/2012
Probability models
6: Rayleigh distribution. Other distributions.
Introduction to generating functions: examples.
Probability generating functions (PGF). PGF of the
Poisson distribution. 20
13
6/10/2012
PGF of uniform and
binomial distributions. Poisson distribution as
limiting case of a binomial distribution. PGF of the
Galton-Watson branching process. PGF of the
Galton-Watson branching process (ctd., application to
photomultipliers). Photomultiplier noise.
22
14
7/11/2012
Photomultiplier noise (ctd.).
Characteristic functions. Moments of a distribution.
Skewness and kurtosis. Mode and median. Propoerties of
characteristic functions.
24
15
8/11/2012
The Central Limit Theorem (CLT). The
Berry-Esseen theorem. Additive and multiplicative
processes. Power-laws from the lognormal distribution.
25
16
13/11/2012
Introduction to discrete-time
stochastic processes. Markov chains. Transient and
persistent states in Markov chains. Transient and
persistent states in the 1D random walk. Transient and
persistent states in higher-dimensional random walks.
Invariant distribution. Time reversal and detailed
balance.
27
17
14/11/2012
Transient and persistent states in the
1D random walk. Transient and persistent states in
higher-dimensional random walks.
29
18
15/11/2012
Invariant distribution. Time reversal
and detailed balance.
30
Part 2: Introduction
to statistical inference
19
19/11/2012
Descriptive statistics.
Sample mean, sample variance, estimate of covariance
and correlation coefficient. Statistics of sample mean
for exponentially distributed data.
32
20
22/11/2012
Limitations of the
standard deviation as a descriptor of the width of a
distribution. Shannon entropy and information
concentration in pdf's.
33
21
27/11/2012
Example of synthesis of
nonparametric estimators in the "German Tank Problem".
The Allan variance for noise processes with infrared
divergences.
Introduction to the Monte Carlo method. Early history
of the Monte Carlo method.
35
22
3/12/2012
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. 37
23
4/12/2012
The structure of a
complete MC program to simulate low-energy electron
scattering. Transformation method and the
trasformation of differential cross sections.
Statistical bootstrap. 39
24
5/12/2012
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.
Maximum likelihood method 1. Point estimators.
Connection with Bayes' theorem. Properties of
estimators. Maximum likelihood method. . 41
25
6/12/2012
Maximum likelihood
method 2. Consistency of the maximum likelihood
estimators. Asymptotic optimality of ML estimators.
Graphical method for the variance of ML estimators.
43
26
10/12/2012
Maximum likelihood
method 3. Bartlett's Identities. Cramer-Rao-Fisher
bound. Variance of ML estimators. Extended maximum
likelihood. Examples.
45
27
11/12/2012
Maximum likelihood
method 4. 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.
Chi-square and least squares fits. 47
28
12/12/2012
Chi-square
distribution. Weighted straight line fits. General
least squares fits. Least squares fitting of binned
data. Nonlinear least squares.
49
29
17/12/2012
Hypothesis test,
significance level. Examples. Critical region.
Construction of test statistics. Neyman-Pearson lemma.
Chi-square and multidimensional confidence intervals.
Chi-square test.
Significance of a signal. Detailed analysis of the
statistical significance of a peak in spectral
estimation. 51