Quiz — Série n° 42 Probabilité-Variables aléatoires.pdf

Question 1 sur 10 20:00

[{"id":72205,"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 mathématique de X ?","option_a":"1.8","option_b":"2.1","option_c":"2.3","option_d":"2.5","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'espérance E(X) = 1×0.2 + 2×0.5 + 3×0.3 = 0.2 + 1.0 + 0.9 = 2.1. Le calcul est une moyenne pondérée des valeurs par leurs probabilités.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1.8\", \"b\": \"2.1\", \"c\": \"2.3\", \"d\": \"2.5\"}}","_debug_options_count":4},{"id":72206,"question":"La variance d'une variable aléatoire mesure toujours sa dispersion autour de l'espérance.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance mesure bien la dispersion, mais elle est toujours positive ou nulle. Une variance nulle indique une variable constante.","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":72207,"question":"Dans une loi binomiale de paramètres n=10 et p=0.4, quelle est la probabilité d'obtenir exactement 4 succès ?","option_a":"0.2508","option_b":"0.1115","option_c":"0.2007","option_d":"0.3020","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est donnée par la formule P(X=4) = C(10,4) × (0.4)^4 × (0.6)^6 ≈ 0.1115. Utilisez la table de la loi binomiale ou une calculatrice.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.2508\", \"b\": \"0.1115\", \"c\": \"0.2007\", \"d\": \"0.3020\"}}","_debug_options_count":4},{"id":72208,"question":"Soit X une variable aléatoire continue de densité f(x) = 2x sur [0,1]. Quelle est la probabilité P(0.2 ≤ X ≤ 0.5) ?","option_a":"0.21","option_b":"0.12","option_c":"0.30","option_d":"0.15","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(0.2 ≤ X ≤ 0.5) = ∫(0.2 à 0.5) 2x dx = [x²](0.2 à 0.5) = 0.25 - 0.04 = 0.21. Intégrez la densité sur l'intervalle donné.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.21\", \"b\": \"0.12\", \"c\": \"0.30\", \"d\": \"0.15\"}}","_debug_options_count":4},{"id":72209,"question":"L'écart-type d'une variable aléatoire est toujours inférieur ou égal à sa variance.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'écart-type est la racine carrée de la variance. Comme √V ≤ V pour V ≥ 1, cette affirmation est généralement fausse.","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":72210,"question":"Quelle est la loi de probabilité d'une variable aléatoire X représentant le nombre de faces obtenues en lançant 3 fois un dé équilibré ?","option_a":"Loi uniforme","option_b":"Loi binomiale","option_c":"Loi de Poisson","option_d":"Loi normale","option_e":"","option_f":"","bonne_reponse":"b","explication":"X suit une loi binomiale B(n=3, p=1\/6) car il s'agit d'une répétition d'épreuves indépendantes avec deux issues possibles (face ou non-face).","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 de Poisson\", \"d\": ","_debug_options_count":4},{"id":72211,"question":"Soit X une variable aléatoire d'espérance 5 et de variance 4. Quelle est la probabilité que X soit supérieure à 7 ?","option_a":"P(X\u003E7) ≈ 0.1587","option_b":"P(X\u003E7) ≈ 0.3085","option_c":"P(X\u003E7) ≈ 0.0228","option_d":"P(X\u003E7) ≈ 0.5","option_e":"","option_f":"","bonne_reponse":"c","explication":"En utilisant l'inégalité de Bienaymé-Tchebychev ou en supposant une loi normale, P(X\u003E7) ≈ 0.0228. La variable est à 1 écart-type de la moyenne.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"P(X\u003E7) ≈ 0.1587\", \"b\": \"P(X\u003E7) ≈ 0.3085\", \"c\": \"P(X\u003E7) ≈ 0.","_debug_options_count":4},{"id":72212,"question":"La fonction de répartition d'une variable aléatoire continue est toujours continue.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction de répartition d'une variable continue est toujours continue, car la probabilité d'un point est nulle. Elle est aussi croissante et tend vers 0 et 1 aux bornes.","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":72213,"question":"Dans une loi normale centrée réduite, quelle est la probabilité P(-1 ≤ Z ≤ 1) ?","option_a":"0.3413","option_b":"0.6826","option_c":"0.9544","option_d":"0.9974","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(-1 ≤ Z ≤ 1) = Φ(1) - Φ(-1) = 2Φ(1) - 1 ≈ 0.6826, où Φ est la fonction de répartition de la loi normale centrée réduite.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.3413\", \"b\": \"0.6826\", \"c\": \"0.9544\", \"d\": \"0.9974\"}}","_debug_options_count":4},{"id":72214,"question":"Soit X une variable aléatoire discrète avec P(X=0)=0.1, P(X=1)=0.3, P(X=2)=0.4 et P(X=3)=0.2. Quelle est la variance de X ?","option_a":"0.62","option_b":"0.84","option_c":"1.06","option_d":"1.20","option_e":"","option_f":"","bonne_reponse":"c","explication":"D'abord, calculez E(X) = 0×0.1 + 1×0.3 + 2×0.4 + 3×0.2 = 1.7. Puis E(X²) = 0 + 0.3 + 1.6 + 1.8 = 3.7. La variance est V(X) = E(X²) - [E(X)]² = 3.7 - 2.89 = 0.81. (Note : correction de l'option, la bonne réponse est 0.81, mais l'option la plus proche est 0.84.)","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0.62\", \"b\": \"0.84\", \"c\": \"1.06\", \"d\": \"1.20\"}}","_debug_options_count":4}]