{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "mit-18-05-s2022-exam1-q5",
    "area": "mathematics",
    "topic": "probability",
    "origin": "external",
    "title": "Joint density, moments, covariance, and correlation",
    "skills": [
      "joint-marginal-conditional",
      "independence",
      "expectation",
      "variance"
    ],
    "priority": "core",
    "difficulty": "hard",
    "estimated_minutes": 24,
    "prerequisites": [
      "basic-algebra"
    ],
    "source_id": "mit-18-05-s2022",
    "locator": {
      "document": "Exam 1",
      "label": "Question 5",
      "page": 6
    },
    "links": {
      "problem": "https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/resources/mit18_05_s22_exam01_pdf/",
      "solution": "https://ocw.mit.edu/courses/18-05-introduction-to-probability-and-statistics-spring-2022/resources/mit18_05_s22_exam01_sol_pdf/"
    },
    "selection_note": "Integrates continuous probability, moments, dependence, and a geometric integration region in one coherent problem.",
    "answer_status": "official-solution"
  }
}