FRM Part I · FRM Exam Part I · Random Variables
A random variable X has E[X] = 4 and E[X^2] = 25. A second variable is defined as W = X^2 - 2X. Treating only these moments as known, and given that X is such that E[X^2] = 25, what is E[W]?
E[W] equals 17. By linearity of expectation, the mean of X squared minus twice X is E[X squared] minus twice E[X], which is 25 minus 8. No further information about the distribution is needed.
- A9
- B17Correct
- C21
- D33
Explanation
E[W] = E[X^2] - 2E[X] = 25 - 2(4) = 17. The value 33 adds instead of subtracts the linear term. The value 9 mistakenly uses (E[X])^2 - 2E[X] ... which is 16 - 8 = 8, or other misuse of moments. Linearity of expectation is the key.
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