Question
Download Solution PDFA fair six-sided die is rolled, with X being the number on the uppermost face. The variance of X is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula
Variance X = σ2x = E[(X – μ)2] = E(X2) – [E(X)]2
Calculation
|
X |
P(X) |
|
1 |
1/6 |
|
2 |
1/6 |
|
3 |
1/6 |
|
4 |
1/6 |
|
5 |
1/6 |
|
6 |
1/6 |
E(X) = μx = μ
⇒ μ = ∑ xifi = ∑xiP(X = xj)
⇒ 1 × 1/6 + 2 × 1/6 + 3 × 1/6 + 4 × 1/6 + 5 × 1/6 + 6 × 1/6
⇒ (1/6)(1 + 2 + 3 + 4 + 5 + 6)
⇒ 7/2
E(X2) = ∑xj2f(xj) = ∑xj2P(X = xj)
⇒ 12 × 1/6 + 22 × 1/6 + ------62 × 1/6
⇒ 1/6(12 + 22 + 32 + 42 + 52 + 62)
⇒ 1/6(1 +4 + 9 + 16 + 25 + 36)
⇒ 91/6
Variance X = E(X2) – [E(X)]2
⇒ 91/6 – (7/2)2
⇒ 91/6 – 49/4
∴ Variance X is 35/12
Last updated on Jul 9, 2025
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