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Unformatted text preview: Copyright c 2010 by Karl Sigman 1 Review of the exponential distribution The exponential distribution has many nice properties; we review them here. A r.v. X has an exponential distribution at rate λ , denoted by X ∼ exp ( λ ), if X is non- negative with c.d.f. F ( x ) = P ( X ≤ x ), tail F ( x ) = P ( X > x ) = 1- F ( x ) and density f ( x ) = F ( x ) given by F ( x ) = 1- e- λx , x ≥ , F ( x ) = e- λx , x ≥ , f ( x ) = λe- λx , x ≥ . It is easily seen that E ( X ) = 1 λ E ( X 2 ) = 2 λ 2 V ar ( X ) = 1 λ 2 . For example, E ( X ) = Z ∞ xf ( x ) dx = Z ∞ xλe- λx dx = Z ∞ F ( x ) dx (integrating the tail method) = Z ∞ e- λx dx = 1 λ . The most important property of the exponential distribution is the memoryless property , P ( X- y > x | X > y ) = P ( X > x ) , for all x ≥ 0 and y ≥ 0, which can also be written as P ( X > x + y ) = P ( X > x ) P ( X > y ) , for all x ≥ 0 and y ≥ . The memoryless property asserts that the residual (remaining) lifetime of X given that its age is at least y has the same distribution as X originally did, and is independent of its age: X forgets its age or past and starts all over again. If X denotes the lifetime of a light bulb, then this 1 property implies that if you find this bulb burning sometime in the future, then its remaining lifetime is the same as a new bulb and is independent of its age. So you could take the bulb and sell it as if it were brand new. Even if you knew, for example, that the bulb had already burned for 3 years, this would be so. We say that X (or its distribution) is memoryless....
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This note was uploaded on 10/25/2010 for the course IEOR E4106 taught by Professor Yao during the Fall '08 term at Columbia.
- Fall '08