## Random variables and random vectors.

### Teaching unit

### Description

Concept of random variable and basic properties (types, decomposition of Lebesgue , characterization, distribution function , punctual probability function and density function , convolution , other transformations ) , random vectors (characterization , transformations, marginal and conditional distributions ) , mathematical expectation , variance , moments, generating and characteristic functions , Laplace transform , sequences of random variables, laws of large numbers , central limit theorem . Notable models of probability distributions of random variables, generating models of random variables.

### Objectives

Know and apply the concepts and basic principles on the probability distribution of random variables and random vectors (characterization, transformations and marginal and conditional distributions ) as well as notable models of distributions of random variables; modeling and solving problems by random variables in communication systems. Interpret results

### Subjects

#### Lecture topics

- Motivation. Random signals and noise.
- Variable one dimensional random.
- Models of probability distributions of random variables.
- Random vectors.
- Mathematics and variance hope.
- Higher order moments.
- Generating and characteristic functions. Laplace transform.
- Sequences of random variables. Laws of Large Numbers. Central Limit Theorem.
- Concluding remarks and applications.
- Solved exercises.

#### Laboratory topics

- Probability calculation II.
- Modeling by random variables.