Explanation 1
Expected value: The probability-weighted balance point.
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Lesson 4 of 18
Calculate a probability-weighted long-run average and use it in expected profit or cost decisions.
Core lesson
Calculate a probability-weighted long-run average and use it in expected profit or cost decisions.
Explanation 1
Expected value: The probability-weighted balance point.
Explanation 2
Expected profit: Probability-weighted profit across outcomes.
Explanation 3
Decision comparison: Compare expected results while retaining risk information.
Terminology
The probability-weighted balance point.
Long-run average demand.
Probability-weighted profit across outcomes.
Average margin under a stocking plan.
Compare expected results while retaining risk information.
Two inventory plans.
Notation and formulas
E(X) = sum_x x p(x)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
See how probability mass determines a long-run balance point.
P(X = x)
0.2
F(x) = P(X <= x)
0.9
E(X)
3
SD(X)
1.0954
| x | P(X=x) | Included in F(4) |
|---|---|---|
| 1 | 0.1 | Yes |
| 2 | 0.2 | Yes |
| 3 | 0.4 | Yes |
| 4 | 0.2 | Yes |
| 5 | 0.1 | No |
Interpretation: Custom PMF, discrete uniform, Bernoulli, or binomial: inspect support and trial structure before choosing.
Worked example
Scenario
A service manager compares staffing costs across possible demand levels.
The lower expected cost may still carry a larger tail risk.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Calculate center, risk, and expected profit from a custom demand PMF.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Expected value is not a guaranteed outcome and may not be a possible value of X.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Expected Value and Long-Run Average by asking about X, support, parameters, assumptions, probability notation, exact versus approximate status, and interpretation. Do not provide a quiz answer before submission or write my final capstone recommendation.
Exit check
Use conceptual hints before submitting. Full formulas, calculations, and interpretation appear afterward.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
Explore additional discrete-distribution explanations from properly licensed or publisher resources.
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