Scientific reasoning

This compulsory course is intended for all students of the Licence SV. It presents the main tools of discrete probabilities which are useful to the biologist for the understanding of random phenomena involving counting variables. The course is placed at a level accessible to a student having as a pre-requisite only the basics of probability calculus approached in high school. The course is based on concrete examples and leads to modeling. 

• A first preliminary part introduces the notion of sets, operations on sets and the simple formalization of propositions.
• The second part introduces the vocabulary of probabilities and covers the elementary calculation of probabilities (tables, trees) and conditional probabilities. The examples are based on concrete situations: calculation of probabilities in a population stratified by age, gender, diagnostic tests (sensitivity/specificity)
• The third part is devoted to the presentation of the main discrete law models: binomial, geometric, fish and their applications. The notion of independent variables is presented in a heuristic way, the objective being to provide tools to compute the expectation and variance of the sum of random variables.
• Some numerical simulations can be presented to illustrate the notion of fluctuation of a random variable or the convergence of the binomial distribution to the normal distribution or the fish distribution.

To provide the basic tools of probability calculus and the use of usual discrete random variables in a context of application to random phenomena from the life sciences.

mathematics at second level, UE HAV109X Methods of calculation

1) Sets

- Notions of element of a set, subset, membership and inclusion, meeting, intersection and complement and know how to use the corresponding basic symbols: ∈,⊂,∩,∪.

- Notation of the sets of numbers N,Z,D,Q.

- Negation of simple propositions (without implication or quantifiers); counterexamples to show that a proposition is false; formulating an implication, a logical equivalence; reciprocal of an implication: simple set examples.

2) Modeling chance: calculating probabilities

- Set (universe) of outcomes. Events. Meeting, intersection, complementary.

- Law (or distribution) of probability. Probability of an event: sum of the probabilities of the outcomes. Relationship P(A∪B)+P(A∩B) =P(A)+P(B).

- Enumeration using tables and trees (product and sum rule).

- Conditional probabilities and independence: conditional probability of an event B knowing an event A of non-zero probability. Notation PA(B).

- Independence of two events and mutual independence

- Partition of the universe (complete systems of events). Formula of total probabilities. Bayes' formula 

- Succession of independent trials, Bernoulli scheme

3) Real random variables 

- Real random variable: modeling of the numerical result of a random experiment; formalization as a function defined on the universe and with real values.

- Law of a random variable. Expectation, variance, standard deviation of a random variable

- Bernoulli test, Bernoulli law

- Binomial law B(n,p) : law of the number of successes. Expression using the binomial coefficients. Binomial coefficients : definition (number of ways to obtain k successes in a Bernoulli scheme of size n), Pascal's triangle.

- Sum of random variables

Linearity of the expectation: E(X+Y) =E(X)+E(Y) and E(aX) =aE(X). (admitted or can be proved for two discrete a.v. on a finite universe)

Additivity relation for independent variables X,Y : 

V(X+Y) =V(X)+V(Y). Relation V(aX) ^=a2V(X).

Application to the expectation and variance of the binomial distribution.

- Other Discrete Laws :  

Uniform law on {1,2,..., n} 

Geometric law (rank of the first success in a sequence of independent Bernoulli tests): expectation (admitted), property of law without memory.

Poisson's law: characteristics and properties (results on the admitted series), approximation of the binomial law by Poisson's law (case of rare events).

 Computational" examples: function pbinom, rbinom, rpois, ppoiss, rgeom, pgeom, illustration/vulgarization of the law of large numbers: convergence of the proportion of an event A on a sample of size n to the probability P(A).

Hourly volumes* :

            CM : 12h

            TD : 21h

            TP:

            Terrain: