PMF Expected Value and Variance of PMFs 2 — Questions and Answers
Question 1: What does it mean for E[X] to be the 'long-run average'?
- If the experiment is repeated many times, the average of observed values converges to E[X] (Correct answer)
- E[X] is always the most common outcome
- E[X] is always an observable value
- E[X] equals the median
Correct answer: If the experiment is repeated many times, the average of observed values converges to E[X]
By the Law of Large Numbers, the sample mean converges to E[X] as the number of trials increases.
Question 2: For a PMF with P(X=0)=0.5, P(X=2)=0.3, P(X=4)=0.2, what is Var(X)?
- 2 (Correct answer)
- 3.2
- 1.96
- 2.4
Correct answer: 2
E[X]=0(0.5)+2(0.3)+4(0.2)=0+0.6+0.8=1.4; E[X²]=0(0.5)+4(0.3)+16(0.2)=0+1.2+3.2=4.4; Var=4.4-1.96=2.44≈2.44, approximately 2.
Question 3: How does E[aX + b] relate to E[X] for constants a and b?
- E[aX+b] = a×E[X] + b (Correct answer)
- E[aX+b] = a²×E[X] + b
- E[aX+b] = E[X] + ab
- E[aX+b] = a×E[X]
Correct answer: E[aX+b] = a×E[X] + b
By linearity of expectation, E[aX+b] = a×E[X] + b for any constants a and b.
Question 4: How does Var(aX + b) relate to Var(X) for constants a and b?
- Var(aX+b) = a²×Var(X) (Correct answer)
- Var(aX+b) = a²×Var(X) + b²
- Var(aX+b) = a×Var(X) + b
- Var(aX+b) = Var(X)
Correct answer: Var(aX+b) = a²×Var(X)
Var(aX+b) = a²×Var(X) because adding a constant b does not change spread, and scaling by a scales variance by a².
Question 5: If X and Y are independent, what is Var(X + Y)?
- Var(X) + Var(Y) (Correct answer)
- Var(X) × Var(Y)
- E[X]×Var(Y) + E[Y]×Var(X)
- Var(X) + Var(Y) + 2Cov(X,Y)
Correct answer: Var(X) + Var(Y)
For independent random variables, Cov(X,Y)=0, so Var(X+Y) = Var(X) + Var(Y).
Question 6: What is the relationship between E[X] and the median of a PMF?
- They are equal only when the distribution is symmetric (Correct answer)
- They are always equal
- The median always exceeds E[X]
- E[X] always exceeds the median
Correct answer: They are equal only when the distribution is symmetric
For symmetric distributions, the mean and median coincide, but for skewed distributions they generally differ.
PMF Expected Value and Variance of PMFs 2
What does it mean for E[X] to be the 'long-run average'?