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. x 3.0 3.38 0.1 s m - = X Z x = m + sz = 3.0 + 0.3 x 1.282 = 3.3846

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

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

e Inverse Exponential F(x) F(a) a 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 e l - a = - ln(0.9)/l

Inverse Exponential a = - ln(0.9)/l = - (-2.3026)/0.005 = 21.07 hrs. 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(x) F(a) x a