Overview
This lecture explains how to find the expected value (mean) of a discrete random variable using a probability distribution, and interprets what the result means.
Discrete Random Variable and Probability Distribution
- A discrete random variable can only take on a finite number of values (e.g., number of workouts per week: 0, 1, 2, 3, 4).
- A valid probability distribution has all probabilities between 0 and 1, and their sum is exactly 1.
Expected Value (Mean) of a Discrete Random Variable
- The expected value is denoted as E(x) or μ (mu), representing the mean outcome over many trials.
- Calculate expected value by multiplying each outcome by its probability and summing the results.
- Formula: E(x) = Σ[x × P(x)]
Example Calculation
- Given probabilities: 0 (0.1), 1 (0.15), 2 (0.4), 3 (0.25), 4 (0.1)
- Compute: 0×0.1 + 1×0.15 + 2×0.4 + 3×0.25 + 4×0.1 = 0 + 0.15 + 0.8 + 0.75 + 0.4 = 2.1
- The expected number of workouts per week is 2.1.
Interpretation of Expected Value
- The expected value (2.1) does not mean you will do exactly 2.1 workouts in a single week.
- Over many weeks, you can expect to average out to the expected value (e.g., about 21 workouts in 10 weeks).
Key Terms & Definitions
- Discrete Random Variable — a variable that takes on specific, separate values.
- Probability Distribution — a table or formula listing all possible values and their probabilities.
- Expected Value (Mean, μ) — the average outcome calculated as the weighted sum of all possible values and their probabilities.
Action Items / Next Steps
- Practice calculating expected values using different probability distributions.
- Review definitions of key terms for upcoming quiz.