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

Exponential Distribution IE 502: Probabilistic Models Jayendran Venkateswaran IE & OR

Exponential Distribution • Random variable X is exponentially distributed with rate λ i.e. X ~ expo(λ), if  − λ λ ≥ x e x 0 = f ( x )  • X has pdf, < 0 x 0   − λ − ≥ x 1 e x 0 = F ( x )  • X has cdf, < 0 x 0  λ φ = = < λ tX ( t ) E [ e ] for t • MGF: λ − t 1 [ ] 1 [ ] = Var X = E X • Mean, and Variance, λ 2 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 of 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 IE502: Probabilistic Models IEOR @ IITBombay