Exponential distribution STAT 587 (Engineering) Iowa State University September 17, 2020
Exponential distribution Probability density function Exponential distribution The random variable X has an exponential distribution with rate parameter λ > 0 if its probability density function is p ( x | λ ) = λe − λx I( x > 0) . We write X ∼ Exp ( λ ) .
Exponential distribution Probability density function - graphically Exponential probability density function Exponential random variables 2.0 Probablity density function, f(x) 1.5 rate 0.5 1.0 1 2 0.5 0.0 0 1 2 3 4 x
Exponential distribution Mean and variance Exponential mean and variance If X ∼ Exp ( λ ) , then � ∞ x λe − λx dx = · · · = 1 E [ X ] = λ 0 and � ∞ � 2 � x − 1 λe − λx dx = · · · = 1 V ar [ X ] = λ 2 . λ 0
Exponential distribution Cumulative distribution function Exponential cumulative distribution function If X ∼ Exp ( λ ) , then its cumulative distribution function is � x λe − λt dt = · · · = 1 − e − λx . F ( x ) = 0 The inverse cumulative distribution function is F − 1 ( p ) = − log(1 − p ) . λ
Exponential distribution Cumulative distribution function - graphically Exponential cumulative distribution function - graphically Exponential random variables 1.00 Cumulative distribution function, F(x) 0.75 rate 0.5 0.50 1 2 0.25 0.00 0 1 2 3 4 x
Exponential distribution Memoryless property Memoryless property Let X ∼ Exp ( λ ) , then P ( X > x + c | X > c ) = P ( X > x ) .
Exponential distribution Parameterization by the scale Parameterization by the scale A common alternative parameterization of the exponential distribution uses the scale β = 1 λ . In this parameterization, we have f ( x ) = 1 β e − x/β I( x > 0) and V ar [ X ] = β 2 . E [ X ] = β and
Exponential distribution Summary Summary Exponential random variable X ∼ Exp ( λ ) , λ > 0 f ( x ) = λe − λx , x > 0 F ( x ) = 1 − e − λx F − 1 ( p ) = − log(1 − p ) λ E [ X ] = 1 λ 1 V ar [ X ] = λ 2
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