The course aims to introduce the basic logical and conceptual methodologies to lead learners to a correct approach to the problems of data analysis. The objectives concern a correct use of formalization and analysis procedures in the application of probability theory and statistics.

Fundamental concepts 

1.1 Probability and random variables  

1.2 Interpretation of probability  

1.3 Probability density functions

1.4 Functions of random variables

1.5 Expectation values

1.6 Error propagation

1.7 Orthogonal transformation of random variables

2 Examples of probability functions 

2.1 Binomial and multinomial distributions

2.2 Poisson distribution

2.3 Uniform distribution

2.4 Exponential disfribution

2.5 Gaussian distribution

2.6 Log-normal distribution

2.7 Chi-square distribution

2.8 Cauchy (Breit-Wigner) distribution

2.9 Landau distribution

3 The Monte Carlo method 

3.1 Uniformly distributed random numbers

3.2 The transformation method

3.3 The acceptance-rejection method

3.4 Applications of the Monte Carlo method

4 Statistical tests 

4.1 Hypotheses, test statistics, significance level, power

4.2 An example with particle selection 

4.3 Choice of the critical region using the Neyman-Pearson lemma

4.4 Constructing a test statistic

4.5 Goodness-of-fit tests

4.6 The significance of an observed signal

4.7 Pearson 's chi2 test

5 General concepts of parameter estimation 

5.1 Samples, estimators, bias

5.2 Estimators for mean, variance, covariance

6 The method of maximum likelihood 

6.1 ML estimators

6.2 Example of an ML estimator: an exponential distribution

6.3 Example of ML estimators

6.4 Variance of ML estimators: analytic method  

6.5 Variance of ML estimators: Monte Carlo method

6.6 Variance of ML estimators: the RCF bound

6.7 Variance of ML estimators: graphical method

6.8 Example of ML with two parameters

6.9 Extended maximum likelihood

6.10 Maximum likelihood with binned data

6.11 Testing goodness-of-fit with maximum likelihood

6.12 Combining measurements with maximum likelihood

6.13 Relationship between ML and Bayesian estimators 

7 The method of least squares

7.1 Connection with maximum likelihood

7.2 Linear least-squares fit

7.3 Least squares fit of a polynomial

7.4 Least squares with binned data

7.5 Testing goodness-of-fit with chi2

7.6 Combining measurements with least squares

8 Statistical errors, confidence intervals and limits 

8.1 The standard deviation as statistical error

8.2 Classical confidence intervals (exact method)

8.3 Confidence interval for a Gaussian distributed estimator

8.4 Confidence interval for the mean of the Poisson distribution

8.5 Confidence interval for correlation coefficient, transformation of parameters

8.6 Confidence intervals using the likelihood function or chi2

8.7 Multidimensional confidence regions

8.8 Limits near a physical boundary

8.9 Upper limit on the mean of Poisson variable with background

9. Time series analysis

9.1 Time random processes

9.2 Relation to probability

9.3 Ensemble correlation functions

9.4 Time averages

9.5 Fourier trasform, discrete Fourier trasform Nyquist frequency and Sampling theorem 

9.6 Power spectral density and its estimation

9.7 Response of linear filters, convolution theorem, aliasing and PSD windowing, correlation and autocorrelation

10. Detection of known signals

10.1 Additive and gaussian noise

10.2 Signal to Noise Ratio (SNR)

10.3 Matched FIlter in time and frequency domain 

There are no prerequisites.

Knowledge and understanding: the student will have to know the fundamental concepts of probability theory and be able to identify the appropriate statistical methodologies in the analysis of experimental data.
Applied knowledge and understanding: the student must be able to apply the methods studied to real problems by providing a correct statistical description of the experimental data and interpreting the results correctly.
Autonomy of judgment: the student must be able to independently evaluate the plausibility of the result of an analysis, both through the comparison between different possible approaches, and through analogical considerations and scientific common sense.
Communication skills: the student will have to acquire a correct scientific language and the ability to explain the statistical characteristics of the analyzed data.
Ability to learn: the student will be able to deepen specific concepts, not presented during the course, on scientific texts.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Teaching

Lectures and classroom exercises.

Attendance

Attendance is strongly recommended.

Course books

Statistical Data Analysis - Glen Cowan - Oxford University Press

Detection of Signals in Noise - RN McDonough, AD Whalen - Academic Press

Assessment

Written test: problems of probability and statistics.

Oral test: questions on the entire program carried out.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Course books

Statistical Data Analysis - Glen Cowan - Oxford University Press

Detection of Signals in Noise - RN McDonough, AD Whalen - Academic Press

Assessment

Written test: problems of probability and statistics.

Oral test: questions on the entire program carried out.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

The student must be able to apply the basic concepts of mathematical analysis.