Introduction to Probability & Statistics

Inverse FunctionsInverse 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

0.1

x = + z

= 3.0 + 0.3 x 1.282

= 3.3846

XZ

Inverse ExponentialExponential Life

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 0.5 1 1.5 2 2.5 3

Time to Fail

Den

sit

y

a

f x e x( )

f(x)F a X a( ) Pr{ }

e dxxa

0

e x a0

1 e a

Inverse Exponential

xF Xe

1 -)(F(x)

x

Suppose we wish to find a such that the probability of a failure is limited to 0.1.

0.1 = 1 -

ln(0.9) = -a

ae

F(x)

x

F(a)

a

Inverse Exponential

a = - ln(0.9)/

Suppose a car battery is governed by an exponential distribution with = 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)/

= - (-2.3026)/0.005

= 21.07 hrs.

F(x)

x

F(a)

a

Inverse Exponential