Degree course: 
Corso di Second cycle degree in PHYSICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Supplementary compulsory subjects
Second semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (80 hours)

Knowledge of the topics covered in Mathematical Analysis 1, in particular, series, differential and integral calculation for functions of a single variable is required.

Final Examination: 

The final exam is a written and will comprise the solution of exercises. The student will also be asked to state and prove one or more theorems among the ones presented in class.
The first part of the final exam that refers to the theorems is to verify the formal understanding of the probability theory.
The exercises, on the other hand, require the student to understand the context of the application and, on the basis of the same, to decide which aleatory variable best describes the context. It also requires that the student understands what probabilistic instrument is best suited to answer specific questions of interest.

Voto Finale

The course's general objective is to provide students with a formal introduction to probability theory, which is the basis for disciplines such as statistics and the study of stochastic processes. The second training objective is more applied and refers to the understanding of the context in which the introduction of probabilistic concepts and random variables is required and, based on that understanding, the student is required to be able to use the most suitable probabilistic instrument and the best random variable/model able to describe the context and the phenomenon of interest.

Historical introduction to probability theory and set theory recalls with reference to probability theory.

• Definition of sample, event and sigma algebra; probability function and its properties (with demonstrations). Examples.

• Review of combinatorics with specific reference to probability calculation.

• Conditional probability; product rule; total probability theorem (with demonstration); Bayes formula (with demonstration); independent events. Examples.

• Random variable; cumulative distribution function and its properties. Examples.

• Discrete random variables: definition; probability mass function and its properties; definition of expected value; expected value of a function of a discrete random variable (with demonstration); properties of expected value (with demonstration); variance and coefficient of variation. Examples.

• Significant examples of discrete random variables: Bernoulli, binomial, Poisson, uniform, geometric and calculation of their expected value and variance (with demonstration); approximation of binomial distribution with Poisson distribution (with demonstration).

• Independent random variables; Independence of functions of random variables or random vectors (with demonstration). Examples.

• Distribution of the sum of discrete random variables (with demonstration); sum of independent binomial and independent Poisson variables (with demonstration). Examples.

• Discrete random variables: monotony of expected value (with demonstration); expected value of the product and variance of the sum of independent random variables (with demonstration); moments; higher order moments imply the existence of lower order moments (with demonstration); Markov and Chebyshev inquealities (with demonstration); Schwarz inequality (with demonstration); covariance, correlation coefficient and its properties (with demonstration).

• Probability generating function: definition and property (with demonstration). Generating function of the sum of independent random variables (with demonstration) with application to the sum of independent binomial and Poisson random variables (with demonstration).

• Continuous random variables: definition; density function and its properties; expected value, variance and moments; continuous k-dimensional random variables; joint distribution function and density; independence; Conditional density function and expected conditional value. Examples.

• Moments generating function and its properties.

• Examples of continuous random variables (uniform, exponential, gamma, chi-square, normal) and their expected value, variance, and moment-generating function (with demonstration). Student's T distribution.

• Distribution a of a function of a discrete and continuous random variable. For the continuous case, the methods of the cumulative distribution function, the method of monotone transformation (with demonstration) and the method of the moment generating function will be analyzed. Examples.

• Distribution of the sum of continuous random variables (with demonstration) and examples; sum of discrete and continuous random variables through moment generating function (with demonstration). Examples.

• Characteristic function: existence (with demonstration); characteristic function of an affine function of a random variable (with demonstration); continuity; moments and characteristic function; independence; characteristic function of the sum of independent random variables; uniqueness of the characteristic function. Examples.

• Convergence: almost sure, in probability and in distribution; law of large numbers; central limit theorem and De Moivre theorem, continuity correction. Examples.

Baldi P. Calcolo delle Probabilità. The McGraw-Hill Companies edizione 2007 o edizione 2011. ISBN: 9788838666957

Front lectures with theory and exercises.

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