Exponential Distribution IE 502: Probabilistic Models Jayendran - - PowerPoint PPT Presentation

Exponential Distribution

IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

IE502: Probabilistic Models IEOR @ IITBombay

Exponential Distribution

• Random variable X is exponentially distributed

with rate λ i.e. X ~ expo(λ), if

• X has pdf,
• X has cdf,
• MGF:
• Mean,

and Variance,    < ≥ =

−

) ( x x e x f

x λ

λ

   < ≥ − =

−

1 ) ( x x e x F

x λ

λ λ λ φ < − = = t t e E t

tX

for ] [ ) (

[ ]

1 E X λ = [ ]

2

1 Var X λ =

IE502: Probabilistic Models IEOR @ IITBombay

Memorylessness Property

• A random variable X is said to be memoryless if

P{ X > t + s | X > t} = P{X > s} , for all t,s ≥ 0

• Only distributions with memoryless property:

Geomteric & Exponential

• Prove X ~expo(λ) is memoryless
• Implication: “The future is independent of the past”

IE502: Probabilistic Models IEOR @ IITBombay

Examples

• 1. Suppose that the time a customer spends in a bank

is exponentially distributed with mean 10 min.

– What is P{customer spends > 5 min in bank} ? – What is P{customer spends > 15 min in bank} ? – What is P{customer spends > 15 min in bank | he has already spent 10 min} ?

• 2. Suppose lifetime (in hrs) of a bulb is expo(0.1). A

person enters a room in which a bulb is already burning.

– If the person desires to work there for 5 hours, what is P{complete the work before bulb burns out} – What is P{complete the work before bulb burns out} if lifetime is not exponential?

IE502: Probabilistic Models IEOR @ IITBombay

Memorylessness: what does it says?

• Reliability: Amount of time a lightbulb has been in

service has no effect on the amount of time left until it fails

• Inter-event times: Amount of time since the arrival of

last bus contains no information about the amount

• f time until the arrival of next bus
• Service times: Amount of remaining service time is

independent of the amount of service time elapsed so far

• Do the above statements seem realistic?

IE502: Probabilistic Models IEOR @ IITBombay

Let’s work on ‘lifetime’

• Suppose X is a random variable representing the

lifetime of a system/ component/ commodity

• Give some real-life examples whose lifetimes can

be modeled by an X such that P{X > t + s | X > t} goes down as t goes up

– That is, system is more likely to fail as time goes on.

• Give some real-life examples whose lifetimes can

be modeled by an X such that P{X > t + s | X > t} goes up as t goes up

– That is, system is less likely to fail as time goes on