Recall your experience when you take an elevator.

Think about usually how long it takes for the elevator to arrive.

Most likely, the experience may be it frequently comes in a short while and once in a while, it may come pretty late.

In another word, if we want to use a random variable to measure the waiting time for elevator to come, we can say that:◦ 1. It must be continuous.◦ 2. Smaller values have larger probability and

larger values have smaller probability.◦ Think about Geometric distribution, is there any

similarities?

Usually, exponential distribution is used to describe the time or distance until some event happens.

It is in the form of:

◦ where x ≥ 0 and μ>0. μ is the mean or expected value.

1( )

x

f x e

In this case,

Then the mean or expected value is

( ) xf x e 1

1

00

0.2

0.4

0.6

0.8

1

x

R(x)

We also use CDF to find probabilities under exponential distribution.

Or

00

0

0

( ) ( ) 1xx

P x x f x dx e

( ) ( ) ( ) ( )b

a

x

a b b a

x

P x x x f x dx P x x P x x

0 0

( ) ( ) ( ) 1 (1 )b a a bb b a

a

x x x xx x x

x

f x dx f x dx f x dx e e e e

E(X)= μ or

Var(X)= μ2 or

1

2

1