Lecture Notes: Random Variable and Expected Value

Definition of a Random Variable

• Random Variable x: Represents the number of workouts in a given week.
• Values x can take: 0, 1, 2, 3, or 4 (finite number of values).
• Type: Discrete random variable because it has a finite number of possible values.

Probability Distribution

• Characteristics:
  • The sum of probabilities is 1:
    • 0.1 + 0.15 + 0.4 + 0.25 + 0.1 = 1
  • All probabilities are non-negative.
• Valid Distribution: Meets the conditions for a valid probability distribution.

Expected Value of a Discrete Random Variable

• Concept:
  • Gives a sense of the average or mean number of workouts in a week.
  • Notation: Often denoted by Greek letter "mu" (μ).

Calculation

• Formula: Weighted sum of outcomes by probabilities:
  • Expected Value (E(x)) = 0 * 0.1 + 1 * 0.15 + 2 * 0.4 + 3 * 0.25 + 4 * 0.1
• Simplified Calculation:
  • 0 * 0.1 = 0
  • 1 * 0.15 = 0.15
  • 2 * 0.4 = 0.8
  • 3 * 0.25 = 0.75
  • 4 * 0.1 = 0.4
• Total:
  • Sum: 0.15 + 0.8 + 0.75 + 0.4 = 2.1

Interpretation

• Expected Value: 2.1 workouts per week.
• Non-integer Value Explanation:
  • Represents an average over multiple weeks, not predicting exact weekly workouts.
  • In 10 weeks, expect approximately 21 workouts.
  • Applicable even when all outcomes are whole numbers.

Conclusion

• An expected value can be a non-integer even for discrete variables.
• Provides useful information for predicting outcomes over time.