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How do you find E(X) for a discrete RV?
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All Flashcards in Topic 4.14
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4.14.18 cards
How do you find E(X) for a discrete RV?
Multiply each value by its probability and add them: E(X) = Σ x·P(X = x). It's the long-run average.
Formula for Var(X) of a discrete RV?
Var(X) = E(X²) − [E(X)]² — the mean of the squares minus the square of the mean. SD = √Var(X).
Does E(X) have to be a value X can take?
No — e.g. the expected number of heads in one flip is 0.5, even though you only ever see 0 or 1.
What is E(aX + b)?
aE(X) + b — the scale multiplies and the shift b is added on.
What is Var(aX + b)?
a²Var(X). The shift b drops out completely; the scale a enters SQUARED.
What is SD(aX + b)?
|a|·SD(X). The standard deviation multiplies by |a| and is unaffected by the shift b.
Why does +b vanish from the variance?
Adding a constant slides every value equally, so the gaps between values (the spread) are unchanged.
On a GDC, how do you get E(X) and SD from a probability table?
Enter values in L1 and probabilities in L2, run 1-Var Stats with L1 as data and L2 as frequencies: x̄ = E(X), σ = SD.
4.14.28 cards
For independent X and Y, what is E(X + Y) and E(X − Y)?
E(X + Y) = E(X) + E(Y); E(X − Y) = E(X) − E(Y). Means take the sign.
For independent X and Y, what is Var(X ± Y)?
Var(X + Y) = Var(X − Y) = Var(X) + Var(Y). Variances ALWAYS add, even for a difference.
Can you add standard deviations to combine spreads?
No — add the VARIANCES (square the SDs), then square-root: SD(X±Y) = √(SD(X)² + SD(Y)²).
Sum of n independent copies of X (mean μ, variance σ²): mean and variance?
Mean = nμ; Variance = nσ² (so SD = σ√n).
Difference between Var(nX) and Var(X₁+…+Xₙ)?
Var(nX) = n²σ² (one copy scaled up); Var(sum of n independent copies) = nσ² (separate items partly cancel).
What is the unbiased estimate of the population mean?
The sample mean x̄ — it's unbiased as is.
What is the unbiased estimate of the population variance?
sₙ₋₁² = Σ(x − x̄)²/(n − 1) — divide by n − 1, the GDC's Sx² (not σx² which uses ÷n).
Relationship between sₙ₋₁² and the biased sₙ²?
sₙ₋₁² = [n/(n − 1)]·sₙ² — scale the biased variance up by n/(n − 1).
Topic 4.14 study notes
Full notes & explanations for E(X), Var(X) & estimators (HL only)
Math AI exam skills
Paper structures, command terms & tips
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