Exponential distribution

Question 1

Exponential distribution
▪ Definition

Answer 1

A continuous random variable X is said to have Exponential distribution with parameter λ>0
X ~ Exp(λ)

Density:
fₓ(x)=
▪ λ*e^(-λx) if x⩾0
▪ 0 if x<0

Fₓ(x)=
▪ 1-e^(-λx) if x⩾0
▪ 0 if x<0

Then
E(X) = 1/λ
Var(X) = 1/λ²

Question 2

Exponential distribution
▪ When use it

Answer 2

▪ time before earthquake
▪ time before customer enter in a shop
▪ time next call
▪ time piece of machinery work

Question 3

Exponential distribution
▪ Properties

Answer 3

a) Loss of memory
b) Convolution of iid exponential random variables
c) Minimum of an independent family of exponential random variables

Question 4

Exponential distribution
▪ Properties
a) Loss of memory

Answer 4

Consider
▪ X ~ Exp(λ)
▪ It s,t>0

P ( X>s+t|X>t ) = P ( X>s )

P ( X>µ ) = 1 - Fx(µ) = e^(-λµ)

Question 5

Exponential distribution
▪ Properties
b) Convolution of iid exponential random variables

Answer 5

Let X₁, …, Xₙ are iid exponential random variables with parameter λ.
Then X₁, …, Xₙ has gamma distribution with parameters n and λ

X₁, …, Xₙ iid~ Exp(λ)
→
X₁, …, Xₙ~ Γ(n, λ)

Question 6

Exponential distribution
▪ Gamma distribution

Answer 6

A continuous random variable X is said to have Gamma distribution with parameter α,λ>0
X ~ Γ(α, λ)

Density:
fₓ(x)=
▪ (λᵃ/Γ(α))xᵃ⁻¹e^(-λx) if x⩾0
▪ 0 if x<0
where
Γ(α)=₀∫⁺∞ xᵃ⁻¹*e^(-x) dx

Then
E(X) = α/λ
Var(X) = α/λ²

Question 7

Exponential distribution
▪ Properties
c) Minimum of an independent family of exponential random variables

Answer 7

Suppose that X₁, …, Xₙ are independent exponential random variables with:
Xᵢ ~ Exp(λᵢ) –> (i = 1, …, n)

The smallest of the Xᵢ has exponential distribution with parameter the sum of the Xᵢ

Proposition
If are independent with:
Xᵢ ~ Exp(λᵢ) –> (i = 1, …, n)
Then
▪ min{X₁, …, Xₙ}~ Exp(λ₁, …, λₙ)
▪ P(Xj = min{X₁, …, Xₙ})=λj/(λ₁+…+λₙ)