Combining variables & estimators
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Flip to reveal answersFor independent X and Y, what is E(X + Y) and E(X − Y)?
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All 8 Flashcards — Combining variables & estimators
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Question
For independent X and Y, what is E(X + Y) and E(X − Y)?
Answer
E(X + Y) = E(X) + E(Y); E(X − Y) = E(X) − E(Y). Means take the sign.
Question
For independent X and Y, what is Var(X ± Y)?
Answer
Var(X + Y) = Var(X − Y) = Var(X) + Var(Y). Variances ALWAYS add, even for a difference.
Question
Can you add standard deviations to combine spreads?
Answer
No — add the VARIANCES (square the SDs), then square-root: SD(X±Y) = √(SD(X)² + SD(Y)²).
Question
Sum of n independent copies of X (mean μ, variance σ²): mean and variance?
Answer
Mean = nμ; Variance = nσ² (so SD = σ√n).
Question
Difference between Var(nX) and Var(X₁+…+Xₙ)?
Answer
Var(nX) = n²σ² (one copy scaled up); Var(sum of n independent copies) = nσ² (separate items partly cancel).
Question
What is the unbiased estimate of the population mean?
Answer
The sample mean x̄ — it's unbiased as is.
Question
What is the unbiased estimate of the population variance?
Answer
sₙ₋₁² = Σ(x − x̄)²/(n − 1) — divide by n − 1, the GDC's Sx² (not σx² which uses ÷n).
Question
Relationship between sₙ₋₁² and the biased sₙ²?
Answer
sₙ₋₁² = [n/(n − 1)]·sₙ² — scale the biased variance up by n/(n − 1).
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Full study notes for Combining variables & estimators
Topic 4.14 hub
E(X), Var(X) & estimators (HL only)
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