Introduction: the notion of distribution in physics. Discrete and continuous distributions, examples.

Basic notions: sample (random) variable. Probability, probability distribution, probability density, conditional probability. Axioms (product and sum rule, normalization and transformation of distributions when continuous variables are transformed). Averages, moments, quantiles.

Examples of distributions: Binomial, Poisson, polynomial distribution. Uniform, exponential, Cauchy and Gauss distribution. Central limit theorem. Dirac delta distribution.

Monte-Carlo method: Generators of pseudo-random numbers. »Hit-and-miss« method. Generation of arbitrary distributions.

Parameter estimation: Confidence level and (inverse) probability, axioms. Bayes theorem and convolution (marginalization). Consistence theorem. Confidence intervals, calibration, Lindley theorem.

Examples: Parameter estimation of exponential and Gaussian distribution. Propagation of uncertainties. Linear models and fitting. Dynamical models and Kalman filter.

Information: Fisher information, Shannon entropy.

# Probability in physics

I. Kuščer, A.Kodre, Matematika v fiziki in tehniki. DMFA, Ljubljana.

W.T.Eadie, D.Drijard, F.E.James, M.Roos, B.Sadoulet, Statistical Methods in Experimental Physics. North-Holland.

Kendall's Advanced Theory of Statistics, Arnold, London: Vol. 1: Distribution Theory. Vol. 2: Classical Inference and The Linear Model. Vol. 3: Bayesian Inference.

To acquaint students with the concepts of probability calculus, with an emphasis on applications in physics.

Knowledge and understanding:

Familiarity with and understanding of the basics of probabilistic conclusions in physics.

Application:

The acquired knowledge is the basis for apprehension of several other subjects (see 16.4).

Reflection:

The students encounter the concept of probability and the corresponding conclusion making for the first time. Through the concept of distribution they connect the probability to known approaches in physics and acquire knowledge that is imperative to absorb the novel physics topics in 2nd and 3rd study year.

Transferable skills:

The course content is the basis for the apprehension of other subjects, e.g. Modern physics 1 and 2, Statistical physics, Quantum mechanics 1 and Physical measurements 1. The acquired knowledge is also of key importance for the interpretation of results in all practicum (laboratory) courses.

Lectures (1 hour per week).

Exercises (1 hour per week), where the students solve interesting problems related to theory.

Written exam is graded.

Grades: 1-5 (negative), 6-10 (positive) (in accordance with UL regulations).

[1] S. Širca, M. Horvat, Računske metode za fizike, DMFA, Ljubljana 2011. (also available as S.

Širca, M. Horvat, Computational Methods for Physicists, Springer-Verlag, Berlin 2013).

[2] O. Gayou, S. Širca et al., Phys. Rev. Lett. 88 (2002) 092301.

[3] J. J. Kelly, S. Širca et al., Phys. Rev. Lett. 95 (2005) 102001.

[4] R. Subedi, S. Širca et al., Science 320 (2008) 1476.

[5] S. Širca, Few-Body Systems 47 (2009) 39.