PMF Expected Value and Variance of PMFs 1 — Questions and Answers
Question 1: How is the expected value E[X] of a discrete random variable X computed from its PMF?
- E[X] = Σ x × P(X=x) over all x (Correct answer)
- E[X] = Σ P(X=x) over all x
- E[X] = max{P(X=x)}
- E[X] = Σ x² × P(X=x) over all x
Correct answer: E[X] = Σ x × P(X=x) over all x
E[X] = Σ x × P(X=x) is the probability-weighted sum of all possible values.
Question 2: For a fair six-sided die, what is E[X]?
- 3.5 (Correct answer)
- 3
- 4
- 21
Correct answer: 3.5
E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5 for a fair die.
Question 3: How is Var(X) computed from a PMF?
- Var(X) = E[X²] - (E[X])² (Correct answer)
- Var(X) = E[X] - (E[X²])
- Var(X) = Σ x² × P(X=x)
- Var(X) = E[(X + E[X])²]
Correct answer: Var(X) = E[X²] - (E[X])²
Var(X) = E[X²] - (E[X])² is the computational formula for variance, using the second moment minus the squared mean.
Question 4: If E[X] = 4 and E[X²] = 20, what is Var(X)?
- 4 (Correct answer)
- 16
- 24
- 80
Correct answer: 4
Var(X) = E[X²] - (E[X])² = 20 - 16 = 4.
Question 5: What is the standard deviation of X if Var(X) = 9?
- 3 (Correct answer)
- 9
- 81
- 4.5
Correct answer: 3
SD(X) = √Var(X) = √9 = 3.
Question 6: For the PMF P(X=1) = 0.4 and P(X=3) = 0.6, what is E[X]?
- 2.2 (Correct answer)
- 2
- 1.8
- 4
Correct answer: 2.2
E[X] = 1×0.4 + 3×0.6 = 0.4 + 1.8 = 2.2.
PMF Expected Value and Variance of PMFs 1
How is the expected value E[X] of a discrete random variable X computed from its PMF?