Chris Morgan, MATH G160csmorgan@purdue.edu

February 27, 2012Lecture 19

Chapter 6.4: Exponential Distribution

1

Exponential Distribution

An exponential random variable is a continuous random variable that measures the life time of some event.

The exponential distribution is used to model the behavior of units that have a constant failure rate (or units that do not degrade with time or wear out).

2

Exponential Distribution

Examples:

- X is the time until an appliance breaks (lifetime).

- X is the time until a light bulb burns out

- X is the time until the next customer arrives at a grocery store.

- X is the mileage you get from one tank of gas.

3

Exponential Distribution

Examples:

X ~ Exp(μ)

PDF:

for x ≥ 0

CDF:

for x ≥ 0 Note: If we want to find P(X > x), then we are finding:

/

)(xe

xf

/1)()( xexFxXP

// )1(1)( xx eexF 4

Exponential PDF

5

Exponential CDF

6

Exponential Expectation and Variance

Expected Value:

Variance:

)(XE

2)( XVar

7

Memoryless Property

An exponential random variable X has the property that “the future is independent of the past.” In other words, the fact that it hasn’t happened yet tells us nothing about how much longer it will take before it does happen.

)()|( tXPsXtsXP

8

Memoryless Property

Proof:

)(

),()|(

sXP

sXtsXPsXtsXP

)(

)(

sXP

tsXP

/

/)(

s

ts

e

e

/te

)( tXP 9

Memoryless Property

Suppose you are waiting for some event to happen. You have already waited 30 seconds, but the event has not happened. What is the probability that you will have to wait an additional 10 seconds?

/10)10()30|40( eXPXXP

10

Exponential Example 1a

Suppose the time X it takes a puppy to run and get a ball follows an exponential distribution with mean of 30 seconds.

State the distribution and Parameters of T:

~ ( 30)X Exp

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Exponential Example 1b

What is the probability that it takes the puppy more than 50 seconds to get the ball?

1889.0)50(1)50( 30

50

eXPXP

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Exponential Example 1c

Assuming independence, what is the probability that it takes the puppy less than 40 seconds to fetch the next 5 balls?

2166.01)]40([

5

30

405

eXP

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Exponential Example 1d

What is the probability that it will take the puppy more than 45 seconds to get the ball given that we know it took the puppy longer than 20 seconds?

)25()20|45( XPXXP

4346.030

25

e

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Exponential Example #2

A trucker drives between a fixed location in Los Angeles and Phoenix. The duration, in hours, of a round trip has an exponential distribution with parameter 1/20. Determine the probability a round trip:

a)Takes at most 15 hours:

b) Takes between 15 and 25 hours:

15

20( 15) 1 0.5276P X e

1859.011)2515( 20

15

20

25

eeXP

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Exponential Example #2b

c) Exceeds 25 hours:

d) Find the probability that a round trip takes at most 40 hours given that it exceed 15 hours:

2865.0)25( 20

25

eXP

)25()15|40( XPXXP

2865.01

7135.0

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Exponential Example #2c

e) Assuming that round-trip durations are independent from one trip to the next, find the probability that exactly two of five round trips take more than 25 hours:

2981.0)7135.0()2865.0(2

5)2( 32

YP

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Exponential Example #2d

f) Find the mean and variance for the amount of time a trip will take:

20)( XE

40020)( 22 XVar

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