Answer: aE[X] + bE[Y] (linearity of expectation).
- A aE[X] + bE[Y] (linearity of expectation)
- B a × b × E[X] × E[Y], treating expectation as multiplicative
- C E[X] + E[Y], dropping the constants a and b entirely
- D a × E[X] × b × E[Y], multiplying all four quantities together
Correct answer: A. aE[X] + bE[Y] (linearity of expectation)
Explanation: Linearity of expectation: E[aX + bY] = aE[X] + bE[Y]. Proof: E[aX+bY] = Σ(ax+by)·P = aΣxP + bΣyP = aE[X]+bE[Y]. Crucially, this holds regardless of whether X and Y are independent or correlated.
Venn diagram of the universal set U with events A and B, showing the intersection (A and B), the parts unique to each event, and the complement region outside both.
Concept context
Chance, events, conditional probability, and Bayes theorem