1. Introduction to Probability & Statistics Exponential Review

2. 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 ( )   

3. 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 

4. 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 

5. 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  ( )