E(X), Var(X) & linear transformations
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Flip to reveal answersHow do you find E(X) for a discrete RV?
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Question
How do you find E(X) for a discrete RV?
Answer
Multiply each value by its probability and add them: E(X) = Σ x·P(X = x). It's the long-run average.
Question
Formula for Var(X) of a discrete RV?
Answer
Var(X) = E(X²) − [E(X)]² — the mean of the squares minus the square of the mean. SD = √Var(X).
Question
Does E(X) have to be a value X can take?
Answer
No — e.g. the expected number of heads in one flip is 0.5, even though you only ever see 0 or 1.
Question
What is E(aX + b)?
Answer
aE(X) + b — the scale multiplies and the shift b is added on.
Question
What is Var(aX + b)?
Answer
a²Var(X). The shift b drops out completely; the scale a enters SQUARED.
Question
What is SD(aX + b)?
Answer
|a|·SD(X). The standard deviation multiplies by |a| and is unaffected by the shift b.
Question
Why does +b vanish from the variance?
Answer
Adding a constant slides every value equally, so the gaps between values (the spread) are unchanged.
Question
On a GDC, how do you get E(X) and SD from a probability table?
Answer
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.
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Topic 4.14 hub
E(X), Var(X) & estimators (HL only)
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