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Introduction to Probability & Statistics Inverse Functions Inverse Functions Actually, we’ve already done this with the normal distribution. Inverse Normal Actually, we’ve already done this with the normal distribution. 0.1 x 3.0 X-m Z= s 3.38 x = m + sz = 3.0 + 0.3 x 1.282 = 3.3846 Inverse Exponential Exponential Life 2.0 1.8 1.6 F ( a ) = Pr{X a} a - x e = 0 dx =-e - x a 0 =1- e - a Density f ( x) = e - x 1.4 1.2 f(x) 1.0 0.8 0.6 0.4 0.2 0.0 0 0.5 1 a 1.5 2 Time to Fail 2.5 3 Inverse Exponential F (x) = 1 - e - X F(x) x Inverse Exponential F(x) Suppose we wish to find a such that the probability of a failure is limited to F(a) 0.1. 0.1 = 1 - e - a ln(0.9) = -a a a = - ln(0.9)/ x Inverse Exponential Suppose a car battery is governed by an exponential distribution with = 0.005. We wish to determine a warranty period such that the probability of a failure is limited to 0.1. a = - ln(0.9)/ = - (-2.3026)/0.005 = 21.07 hrs. F(x) F(a) x a