Variance of an Indicator Mathematics for Computer Science MIT 6.042J/18.062J I an indicator with E[I]=p: Var[I] :: = E[(I − p) 2 ] Variance = E[I 2 ] − 2pE[I] + p 2 = E[I] − 2p ⋅ p + p 2 = 2 2 = p-2p + p pq variance.1 variance.2 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 Calculating Variance Calculating Variance Var[aR + b] = a 2 Var[R] Var[aR + b] = a 2 Var[R] Var[R] = E[R 2 ] -(E[R]) 2 Var[R] = E[R 2 ] -E 2 [R] simple proofs applying linearity of E[] to the def of Var[] Albert R Meyer, May 10, 2013 variance.3 Albert R Meyer, May 10, 2013 variance.4 1
proof of 2 nd Variance Formula Space Station Mir Var[R] ::= E[(R - μ) 2 ] Destructs with probability p = E[R 2 - 2μ R + μ 2 ] in any given hour = E[R 2 ] − 2μ · E[R] + E[μ 2 ] E[F] = 1 / p (Mean Time to Fail) = E[R 2 ] − 2μ · μ + μ 2 = E[R 2 ] − μ 2 Var[F] = ? = E[R 2 ] − E 2 [R] variance.5 variance.6 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 Variance of Time to Failure Variance of Time to Failure ∞ E[F 2 ] :: = ∑ k 2 ⋅ Pr[F 2 =k 2 ] Pr[F = k] = q k − 1 p k=1 ∞ 2 ] − E 2 [F] = ∑ k 2 ⋅ Pr[F =k] Var[F] = E[F F = 1, 2, 3,…, k ,... k=1 ∞ p = ∑ k 2 q k F 2 = 1, 4, 9,…, k 2 ,… q k� k=0 = � = has closed form Albert R Meyer, May 10, 2013 variance.7 Albert R Meyer, May 10, 2013 variance.8 2
Variance of Time to Failure Conditional time to failure total expectation E[F 2 ] = lemma: For F = time to approach: g : R → R failure, , E[F 2 |F =1] ⋅ Pr[F =1] E[ g(F) |F > n] +E[F 2 |F > 1] ⋅ Pr[F > 1] variance.9 variance.10 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 Conditional time to failure Variance of Time to Failure total expectation E[F 2 ] = lemma: For F = time to approach: g : R → R failure, , E[F 2 |F =1] ⋅ Pr[F =1] E[ g(F) |F > n] = E[ g(F + n)] +E[F 2 |F > 1] ⋅ Pr[F > 1] Corollary: E[F 2 |F > 1] =E[(F + 1) 2 ] Albert R Meyer, May 10, 2013 variance.11 Albert R Meyer, May 10, 2013 variance.12 3
Variance of Time to Failure Variance of Time to Failure total expectation total expectation E[F 2 ] = E[F 2 ] = approach: approach: ⋅ ⋅ 1 p 1 p +E[F 2 |F > 1] ⋅ Pr[F > 1] +E[(F +1) 2 ] ⋅ q variance.13 variance.14 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 Variance of Time to Failure Mean Time to Failure total expectation E[F 2 ] = Var[F] = 1 1 approach: p -1 p ⋅ 1 p Mir1: + (E[F 2 ] + 2/ p + 1) p = 10 -4 , E[F] = 10 4 , σ < 10 4 ⋅ q so by Chebyshev Pr[lasts ≥ 4 10 4 hours] ≤ 1/4 now solve for E[F 2 ] Albert R Meyer, May 10, 2013 variance.15 Albert R Meyer, May 10, 2013 variance.18 4
Calculating Variance Mean Time to Failure 1 1 Pairwise Independent Additivity Var[F] = -1 p Var[R 1 + R 2 + � + R n ] p Mir1: = Var[R 1 ] + Var[R 2 ] + � + Var[R n ] p = 10 -4 , E[F] = 10 4 , σ < 10 4 providing R 1 ,R 2 ,…,R n are pairwise independent so by Chebyshev Pr[lasts ≥ 4.6 years ] ≤ 1/4 again, a simple proof applying linearity of E[] to the def of Var[] variance.19 variance.20 Albert R Meyer, May 10, 2013 Albert R Meyer, May 10, 2013 5
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