1. Introduction to Probability & Statistics Inverse Functions

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

3. 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
= x 1.282 =

4. 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 

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

6. 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

7. 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 = We wish to determine a warranty period such that the probability of a failure is limited to 0.1.
a = - ln(0.9)/l
= - ( )/0.005
= hrs.
F(x)
F(a) x a