Travaux dirigés (TD) - Mathématiques

This document contains practical exercises covering multiple statistical concepts including probability distributions, estimation techniques, calculation of confidence intervals, hypothesis testing, and statistical properties (bias and efficiency). The methodology emphasizes working with real-world data through sampling, and the purpose is to answer questions using inferential statistics such as means, proportions, and standard deviations. Findings include applying theoretical models (e.g., Binomial and Normal distributions) and performing estimations of unknown parameters based on sample d...

This document includes a set of exercises focusing on probability distributions, statistical inference, and real-world applications. It introduces problems involving the calculation of probability distributions, cumulative distribution functions, expected values, and variances. Practical scenarios such as machine precision in production lines and demand forecasting using normal distributions are analyzed. The document aims to strengthen statistical analytical skills through applied problems.

This document consists of a comprehensive series of exercises focused on Boolean algebra and logic circuits. Key tasks include simplifying logical expressions using Boolean algebra, Karnaugh maps, and NAND gates. Additional exercises involve constructing truth tables, transforming expressions into canonical forms, and designing circuit diagrams. The document emphasizes the efficient implementation of logical functions with a minimal number of gates.

This document provides a comprehensive exercise set focused on polynomial interpolation and polynomial approximation concepts in numerical analysis. The first section covers practical computations of interpolating polynomials, error analysis, and applications in least-squares approximation. The second section delves deeper into approaches involving Newton's polynomial basis with derivations for interpolated points and error metrics. Real-world scenarios, such as vehicle braking distance prediction, are used to contextualize theoretical methods. The exercises emphasize precision and interdis...

This document contains a series of exercises focused on the application of linear programming models and graphical resolution methods to maximize profitability and optimize operations. Problems include production planning for factories (e.g., products and raquettes), petroleum refinery optimization for maximum margins, and workshop scheduling to efficiently allocate time and resources. Each exercise specifies constraints like production limits, resource availability, and profitability to develop and solve linear programming models.

This document provides a set of four exercises focused on hypothesis testing involving real-world applications. Exercise 1 evaluates whether the average time programmers need to complete a task exceeds two hours using a normal distribution model. Exercise 2 tests if the mean learning time for executives exceeds a specific threshold using a t-test. Exercise 3 assesses the variance of light bulb lifespans to determine if it meets client specifications based on a chi-squared test. Exercise 4 evaluates if a pharmaceutical treatment's success rate differs from the claimed percentage through a pr...

This document addresses Gaussian random vectors, focusing on analyzing the properties of a trivariate Gaussian vector X = (X1, X2, X3). Methodologies include determining the distribution of a bivariate Gaussian pair (X1, X2) through affine transformations and verifying the independence of X3 from X1 and X2 based on covariance properties and Gaussian independence rules. Covariance matrix manipulation is used to derive key results. The findings confirm the independence among specified variables using their uncorrelated nature derived from the covariance matrix.

This document provides detailed solutions to an exercise concerning Gaussian vectors. It determines the covariance matrix of a centered Gaussian vector and derives its characteristic function. It also establishes the necessary and sufficient conditions for the vector to have a density, presenting this density explicitly under the invertibility condition of the covariance matrix.

This document is a series of exercises designed to improve proficiency with Laplace transform techniques applied in differential equations. It covers calculating Laplace transforms, finding original functions, solving specific differential equations, and dealing with rational functions through decomposition and convolution. The focus is on practical application of Laplace transforms in solving mathematical and engineering problems.

The document presents the solution to a probability exercise involving a random variable X with a given generating function. The determination of the constant k is first obtained from normalization of the generating function, followed by deriving the probability distribution of X based on its polynomial properties, limiting its outcomes to a discrete set of values {0, 2, 4, 6}. Finally, the expectation and variance of X are computed using the connection between the generating function and the moments of the random variable.

This document comprises an exercise sheet for fourth-year students in data science and mathematics, focusing on foundational probability tools. It introduces various probability problems, such as calculating probabilities in events and determining the distribution and properties of random variables. The problems explore real-world scenarios, including a tennis tournament and abstract mathematical problems like finding probability distributions and expectations. Techniques involve interpreting event probabilities, working with random variables, and using probability density functions.

This document provides a series of exercises related to statistical inference and probability theory. It introduces problems involving probability distributions (e.g., binomial and normal), expectations, variances, and cumulative distribution functions. Practical applications include quality control in manufacturing and market demand estimation. Key methodologies involve mathematical derivations, distribution analysis, and probabilistic decision-making.