This course is intended to provide the student with the fundamental concepts of the probability calculus.
Pedagogical Objectives: The learning purposes of this course are:
- Establishing firm theoretical bases allowing giving rigorous solutions for problems already solved purely intuitively.
- Providing the under graduating students with profound understanding of the probability calculus in order to get them more able to follow the graduating courses.
- Allowing the student to reason in a probabilistic way in face of problems and applications in relevant fields.
- Probability model: Fundamental set, tribe of events, probability law, probability axioms, Poincaré formula, equiprobability.
- Independent events, conditional probabilities, total probability formula, causal probability formula (Bayes theorem).
- Discrete random variable, continuous random variable, couple of random variables, random vector, independent random variables, sum of random variables, product of random variables.
- Characteristics of a random variable: probability law, density function, expectation, variance, standard deviation, variation coefficient, moments, distribution function, characteristic function, generating function.
- Characteristics of a couple of random variables and random vector: probability law, marginal laws, conditional laws, expectation, variance, covariance and linear correlation coefficient between two random variables, variance-covariance matrix of a random vector, probability law of the sum and the product of independent random variables.
- Usual probability distribution: Bernoulli distribution, binomial distribution, hypergeometric distribution, discrete uniform distribution, Poisson distribution, geometric distribution, negative binomial distribution, multinomial distribution, continuous uniform distribution, exponential distribution, normal distribution, Gamma distribution, Beta distribution, Student distribution, Chi-square distribution, Fisher-snedecor distribution, Cauchy distribution, multinormal distribution..
- Elementary limit theorems, convergence in law, approximations of usual laws.