Exponential distribution STAT 587 (Engineering) Iowa State - - PowerPoint PPT Presentation

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

0.0 0.5 1.0 1.5 2.0 1 2 3 4

x Probablity density function, f(x) rate

0.5 1 2

Exponential random variables

Exponential distribution Mean and variance

Exponential mean and variance

If X ∼ Exp(λ), then E[X] = ∞ x λe−λxdx = · · · = 1 λ and V ar[X] = ∞

• x − 1

λ 2 λe−λxdx = · · · = 1 λ2 .

Exponential distribution Cumulative distribution function

Exponential cumulative distribution function

If X ∼ Exp(λ), then its cumulative distribution function is F(x) = x λe−λtdt = · · · = 1 − e−λx. The inverse cumulative distribution function is F −1(p) = − log(1 − p) λ .

Exponential distribution Cumulative distribution function - graphically

Exponential cumulative distribution function - graphically

0.00 0.25 0.50 0.75 1.00 1 2 3 4

x Cumulative distribution function, F(x) rate

0.5 1 2

Exponential random variables

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 E[X] = β and V ar[X] = β2.

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

V ar[X] =

1 λ2