lesson
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Part 1: Basics of probability theory and probability
models
1
3/10/2011
Foundations
of probability. Basic concepts. Stirling's
Approximation.
2
2
5/10/2011
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
6/10/2011
Example:
gambler's ruin and probabilistic diffusion models.
First Borel-Cantelli lemma. Second Borel-Cantelli
lemma.
6
4
10/10/2011
Discrete
and continuous random variables. 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
5
12/10/2011
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
6
13/10/2011
Some
important probability models: uniform distribution,
binomial distribution and multinomial distribution.
The Poisson distribution, the exponential
distribution. Paralyzable and non-paralyzable
detectors (application of the exponential
distribution).
12
7
17/10/2011
Probability
models 2: The De Moivre-Laplace theorem and the
Gaussian distribution. Transformations of random
variables. Approximate transformation of random
variables: error propagation. Linear (orthogonal)
transformation of random variables. The multivariate
normal distribution. Lognormal distribution.
14
8
19/10/2011
Probability
models 3: Gamma distribution. Cauchy
distribution. Landau distribution.
Introduction to generating function: examples.
Probability generating functions (PGF).PGF of the
Poisson distribution.
16
9
20/10/2011
PGF
of uniform and binomial distributions. Poisson
distribution as limiting case of a binomial
distribution. PGF of the Galton-Watson branching
process (application to photomultipliers).
Photomultiplier noise.
18
10
24/10/2011
Characteristic
functions. Moments of a distribution. Skewness and
kurtosis. Mode and median. The Central Limit Theorem
(CLT).
20
11
26/10/2011
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. Transient and
persistent states in the 1D random walk.
22
12
27/10/2011
Transient
and persistent states in higher-dimensional random
walks. Invariant distribution. Time reversal and
detailed balance.
Branching processes in biophysics. The Luria-DelbrŸck
experiment. 24
Part 2: Introduction
to statistical inference
13
7/11/2011
Descriptive
statistics. Sample mean, sample variance, estimate of
covariance and correlation coefficient. Statistics of
sample mean for exponentially distributed data.
26
14
9/11/2011
Example:
the "German Tank Problem".
The Monte Carlo method. Pseudorandom numbers.
Uniformly distributed pseudorandom numbers. 28
15
10/11/2011
Transformation
method. Acceptance-rejection method. Examples:
generation of angles in the e+e- -> mu+mu-
scattering; generation of angles in the Bhabha
scattering.
30
16
14/11/2011
The
structure of a complete MC program to simulate
low-energy electron scattering.
32
17
16/11/2011
Statistical
bootstrap.
34
18
17/11/2011
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.
36
19
21/11/2011
Maximum likelihood method 1. Point estimators.
Connection with Bayes' theorem. Properties of
estimators. Maximum likelihood method. Consistency of
the maximum likelihood estimators.
38
20
23/11/2011
Maximum
likelihood method 2. Variance of ML estimators. The
CramŽr-Rao-Fisher bound. Asymptotic optimality of ML
estimators. Graphical method for the variance of ML
estimators.
40
21
24/11/2011
Maximum
likelihood method 3. Extended maximum likelihood.
Examples. Introduction to ML with binned data. Example
with two channels.
42
22
30/11/2011
Other
examples of ML with binned data: decay rate in
radioactive decay; exponent of power-law.
Chi-square and least squares fits. Chi-square
distribution.
44
23
1/12/2011
Weighted
straight line fits. General least squares fits. Least
squares fitting of binned data. Nonlinear least
squares.
46
24
5/12/2011
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.
Discussion of possible exam topics.
48