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Introduction to Probability & Statistics Inverse Functions

Introduction to Probability & Statistics Inverse Functions. Inverse Functions. Actually, we’ve already done this with the normal distribution. 0.1. x. -. m. X. =. Z. 3.0. 3.38. s. Inverse Normal. Actually, we’ve already done this with the normal distribution. x = m + s z

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Introduction to Probability & Statistics Inverse Functions

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  1. Introduction to Probability & StatisticsInverse Functions

  2. Inverse Functions • Actually, we’ve already done this with the normal distribution.

  3. 0.1 x - m X = Z 3.0 3.38 s Inverse Normal • Actually, we’ve already done this with the normal distribution. x = m + sz = 3.0 + 0.3 x 1.282 = 3.3846

  4. f ( x )   e   x F ( a )  Pr{ X  a } a   e   x dx  0   e   x a 0  1  e   a Inverse Exponential Exponential Life 2.0 1.8 1.6 1.4 1.2 f(x) Density 1.0 0.8 0.6 0.4 0.2 0.0 0 0.5 1 1.5 2 2.5 3 a Time to Fail

  5. F(x) - l X e x Inverse Exponential F ( x ) = 1 -

  6. F(x) F(a) x - l a e a Inverse Exponential Suppose we wish to find a such that the probability of a failure is limited to 0.1. 0.1 = 1 - ln(0.9) = -la a = - ln(0.9)/l

  7. Inverse Exponential F(x) Suppose a car battery is governed by an exponential distribution with l = 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)/l = - (-2.3026)/0.005 = 21.07 hrs. F(a) x a

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