Quiz — 32-variables-aleatoires.pdf

Question 1 sur 10 20:00

[{"id":9893,"question":"Soit X une variable aléatoire discrète prenant les valeurs 1, 2 et 3 avec les probabilités respectives 0.2, 0.5 et 0.3. Quelle est l'espérance de X ?","option_a":"1.8","option_b":"2.1","option_c":"2.5","option_d":"3.0","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance E(X) = 1×0.2 + 2×0.5 + 3×0.3 = 0.2 + 1.0 + 0.9 = 2.1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1.8\", \"b\": \"2.1\", \"c\": \"2.5\", \"d\": \"3.0\"}}","_debug_options_count":4},{"id":9894,"question":"La variance d'une variable aléatoire est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance est définie comme l'espérance du carré de l'écart à la moyenne, donc elle est toujours positive ou nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9895,"question":"Quelle loi de probabilité modélise le nombre de succès dans n épreuves indépendantes de Bernoulli ?","option_a":"Loi uniforme","option_b":"Loi binomiale","option_c":"Loi exponentielle","option_d":"Loi normale","option_e":"","option_f":"","bonne_reponse":"b","explication":"La loi binomiale B(n,p) modélise le nombre de succès dans n épreuves indépendantes avec une probabilité p de succès à chaque épreuve.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Loi uniforme\", \"b\": \"Loi binomiale\", \"c\": \"Loi exponentielle\", \"d","_debug_options_count":4},{"id":9896,"question":"Soit X une variable aléatoire continue de densité f(x) = 2x sur [0,1]. Quelle est la probabilité P(X ≤ 0.5) ?","option_a":"0.125","option_b":"0.25","option_c":"0.5","option_d":"0.75","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(X ≤ 0.5) = intégrale de 0 à 0.5 de 2x dx = [x²] de 0 à 0.5 = 0.25.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.125\", \"b\": \"0.25\", \"c\": \"0.5\", \"d\": \"0.75\"}}","_debug_options_count":4},{"id":9897,"question":"Deux variables aléatoires X et Y sont indépendantes si et seulement si leur covariance est nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La covariance nulle est une condition nécessaire mais pas suffisante pour l'indépendance. Il existe des variables non indépendantes avec une covariance nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9898,"question":"Quelle est la variance d'une variable aléatoire suivant une loi binomiale B(n,p) ?","option_a":"n·p","option_b":"n·p·(1-p)","option_c":"n²·p","option_d":"p·(1-p)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La variance d'une loi binomiale B(n,p) est n·p·(1-p).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"n·p\", \"b\": \"n·p·(1-p)\", \"c\": \"n²·p\", \"d\": \"p·(1-p)\"}}","_debug_options_count":4},{"id":9899,"question":"Soit X une variable aléatoire d'espérance 5 et de variance 4. Quelle est l'espérance de Y = 2X + 3 ?","option_a":"8","option_b":"10","option_c":"13","option_d":"16","option_e":"","option_f":"","bonne_reponse":"c","explication":"E(Y) = 2·E(X) + 3 = 2×5 + 3 = 13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"8\", \"b\": \"10\", \"c\": \"13\", \"d\": \"16\"}}","_debug_options_count":4},{"id":9900,"question":"La loi exponentielle est utilisée pour modéliser des durées de vie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La loi exponentielle est couramment utilisée pour modéliser des temps d'attente ou des durées de vie sans vieillissement.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9901,"question":"Quelle est la probabilité qu'une variable aléatoire X suivant une loi normale centrée réduite prenne une valeur inférieure à -1 ?","option_a":"0.1587","option_b":"0.3413","option_c":"0.5","option_d":"0.8413","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée réduite, P(X ≤ -1) ≈ 0.1587 (valeur lue dans la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.1587\", \"b\": \"0.3413\", \"c\": \"0.5\", \"d\": \"0.8413\"}}","_debug_options_count":4},{"id":9902,"question":"Soit X une variable aléatoire d'espérance μ et de variance σ². Quelle est la variance de Z = (X - μ)\/σ ?","option_a":"0","option_b":"1","option_c":"σ²","option_d":"μ²","option_e":"","option_f":"","bonne_reponse":"b","explication":"Z est la variable centrée réduite. Sa variance est toujours égale à 1, quelle que soit la loi de X.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"σ²\", \"d\": \"μ²\"}}","_debug_options_count":4}]