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Code source wiki de Mathematics - Probabilities

Modifié par Anaïs Stirnemann le 11/06/2025 - 17:05
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3 (% style="color:#000080" %)**Mathematics - Probability theory**
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5 (% style="color:#000080" %)**5JUCSN05**
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7 **ECTS**
8 )))|(% class="highlight-#fff0b3" data-highlight-colour="#fff0b3" style="text-align:center" title="Couleur d'arrière-plan : Jaune clair 100 %" %)(((
9 **3**
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11 **SEMESTER**
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13 **5**
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16 lectures
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18 classes / seminars
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22 integrative teaching
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24 independent work
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35 30h
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38 **Language used**
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46 **Course supervisor(s)**
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49 Coralie FRITSCH & Takéo TAKAHASHI
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52 **Key words**
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54 probability, measure theory, Lebesgue integral, random variable, law of large numbers, central limit theorem, Gaussian vectors
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57 **Prerequisites**
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59 basic knowledge of calculus, linear algebra, Riemann integral and elementary set theory
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62 **Overall objective**
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65 The objective of the course is to master classical notions in probability theory, in measure theory, and to know how to compute the law of some random variables
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68 **Course content and organisation**
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71 This course is concerned with classical notions in probability and measure theory, which include the following topics:
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73 ~1. Probability spaces, random variable and expectation.
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75 2. Discrete and absolutely continuous distributions, classical distributions, Law of the unconscious statistician.
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77 3. Characterization of laws through the cumulative distribution and characteristic functions. Computation of the law of a random variable.
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79 4. Random vectors, marginal laws, Fubini's theorem, Change of variable
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81 5. Fatou's Lemma, Monotone convergence theorem, Dominated convergence theorem
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83 6. Lp spaces, moments, variance, covariance, linear regression
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85 7. Independance and Convolution
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87 8. Different notions of convergence of random variables : in probability, in distribution, almost sure, in Lp. Borel Cantelli Lemma.
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89 9. Gaussian vectors
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91 Students will also be provided with the basic tools to numerically simulate random variables in Python and Matlab.
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94 **Skills**
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97 (% style="color:#0070c0" %)**Levels**
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99 (% style="color:#0070c0" %)**Description and operational verbs**
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102 **Know**
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104 The fundamental aspects of measure theory and the basic tools for computation of probability distributions
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107 **Understand**
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109 To understand the logic of probability theory and the way it applies to concrete calculous
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112 **Apply(% style="color:#0070c0" %) (%%)**
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114 To deploy calculus methods to resolve problems in probability, choosing among different tools to describe random behaviours
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117 **Analyse(% style="color:#0070c0" %) (%%)**
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119 To detect basic properties of random phenomena and to devise a mathematical strategy to analyse them
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122 **Summarise**
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124 To elaborate a well structured discussion about topics on random phenomena
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127 **Assess**
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129 To detect aberration in probability results and assess the validity of different property concerning random phenomena, as independance, convergence or integrability
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132 (% class="kYiyGk sc-14kwckt-9" %)**Compliance with the United Nations Sustainable Development Goals**
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318 **Evalution methods**
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