Explanation 1
Linear transformation: Y=aX+b shifts and rescales X.
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Lesson 17 of 18
Transform costs and combine uncertain quantities using covariance-aware mean and variance rules.
Advanced / Optional
Transform costs and combine uncertain quantities using covariance-aware mean and variance rules.
Explanation 1
Linear transformation: Y=aX+b shifts and rescales X.
Explanation 2
Aggregate mean: Means add regardless of dependence.
Explanation 3
Covariance: Joint movement that changes aggregate variance.
Terminology
Y=aX+b shifts and rescales X.
Demand converted to cost.
Means add regardless of dependence.
Regional demand total.
Joint movement that changes aggregate variance.
Weather-linked regional demand.
Notation and formulas
E(aX+b)=aE(X)+b; Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)
Define X, its support, parameters, units, and assumptions before substituting values.
Interactive Mission
Transform center and spread under Y=aX+b.
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.
Transformed mean: 16; transformed SD: 2.1909; two-variable variance with covariance: 18.4. Standard deviations were not added directly.
Interactive Mission
Add means and compare variance assumptions.
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.
Transformed mean: 16; transformed SD: 2.1909; two-variable variance with covariance: 18.4. Standard deviations were not added directly.
Interactive Mission
See how positive or negative co-movement changes aggregate risk.
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.
Transformed mean: 16; transformed SD: 2.1909; two-variable variance with covariance: 18.4. Standard deviations were not added directly.
Worked example
Scenario
Two regions share weather and promotion effects.
Positive covariance reduces the apparent diversification benefit.
R connection
Run and edit the original script, inspect intermediate output, and keep exact and approximate results clearly labelled.
Live R Lab
Calculate transformed and aggregate risk under explicit dependence assumptions.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
Never add standard deviations directly. Add variances only under the stated covariance or independence assumptions.
Reflection
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
Assistant
Coach me through Transformations and Sums of Random Variables 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 4. Answered 0/4. Passing score: 70%.
Resource
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