Introduction to Probability & Statistics Exponential Review

Example Let X = lifetime of a machine where the life is governed by the exponential distribution. determine the probability that the machine fails within a given time period a. , x > 0,  > 0 f x e ( )   

Example  f x e ( )   F a X ( ) Pr{ }     e dx   e   1 e a 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 

Complementary  F a X ( ) Pr{ }     e dx  e Exponential Life Suppose we wish to know the probability that the machine will last at least a hrs? 2.0 1.8 1.6 1.4 1.2 f(x) Density 1.0 0.8 0.6 F a X ( ) Pr{ }   0.4 0.2 0.0     e dx x  a 0.5 1 1.5 2 2.5 3 a Time to Fail   e a 

Example Suppose for the same exponential distribution, we know the probability that the machine will last at least a more hrs given that it has already lasted c hrs. a c c+a Pr{X > a + c | X > c} = Pr{X > a + c  X > c} / Pr{X > c} = Pr{X > a + c} / Pr{X > c}    e c a  ( )