These quantities describe the “spread” around the expected value μ = E(X).
V(X) = E((X−μ)²) = E(X²) − {E(X)}² (variance)
σ(X) = √V(X) (standard deviation)
Example ① Find the variance and standard deviation of a die roll.
Solution:E(X²) = (1+4+9+16+25+36)/6 = 91/6.
V(X) = 91/6 − (3.5)² = 91/6 − 49/4 = (182−147)/12 = 35/12 ≈ 2.917
σ(X) = √(35/12) ≈ 1.708
Solution:E(X²) = (1+4+9+16+25+36)/6 = 91/6.
V(X) = 91/6 − (3.5)² = 91/6 − 49/4 = (182−147)/12 = 35/12 ≈ 2.917
σ(X) = √(35/12) ≈ 1.708
Key Point
- V(aX+b) = a²·V(X) (a constant does not affect variance)
Practice Problems
- Find the variance when X=0,1,2 have probabilities 1/4, 1/2, 1/4, respectively.
Solutions and Explanations
- Solution:1/2
E(X)=1, E(X²)=0·1/4+1·1/2+4·1/4=3/2. V=3/2−1=1/2.