توضیح 1
Linear transformation: Y=aX+b shifts and rescales X.
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درس 17 از 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.
توضیح 1
Linear transformation: Y=aX+b shifts and rescales X.
توضیح 2
Aggregate mean: Means add regardless of dependence.
توضیح 3
Covariance: Joint movement that changes aggregate variance.
اصطلاحات
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.
نمادگذاری و فرمولها
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.
فعالیت تعاملی
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.
فعالیت تعاملی
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.
فعالیت تعاملی
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.
مثال حلشده
سناریو
Two regions share weather and promotion effects.
Positive covariance reduces the apparent diversification benefit.
ارتباط با R
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.
بازاندیشی
Name the random variable, support, parameters, units, dependence assumptions, and one limitation.
دستیار
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.
ارزیابی پایانی
منبع
Explore additional discrete-distribution explanations from properly licensed or publisher resources.
دریافتمنبع
Download an original worksheet, formula sheet, or capstone tool for this lesson.
دریافت