Question

Compute the quantile function of the exponential distribution with parameter A. Find its median (the 50th percentile)

Answer 1

In probability theory and statistics, the exponential distribution (known as the negative exponential distribution) is the probability distribution of the time between events in a poison point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Exponential

Probability density function 1.6 A0.5 1.4 1.2 1.5 1.0 0.8 0.6 0.4 0.2 0.00 1 2 4 Ln X

Cumulative distribution function 1.0 0.8 0.6 0.4 0.5 =1 0.2 =1.5 0.00 1 4 2 Ln X (XX)d

• Exponential distribution is closed under scaling by a positive factor. If X ~ Exp(λ) then kX ~ Exp(λ/k).
• If X ~ Exp(1/2) then X ∼ χ2
  2, i.e. X has a chi-squared distribution with 2 degrees of freedom. Hence:

{\displaystyle \operatorname {Exp} (\lambda )={\frac {1}{2\lambda }}\operatorname {Exp} \left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum _{i=1}^{n}\operatorname {Exp} (\lambda )\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}}${\displaystyle \operatorname {Exp} (\lambda )={\frac {1}{2\lambda }}\operatorname {Exp} \left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum _{i=1}^{n}\operatorname {Exp} (\lambda )\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}}$

• If X_i ~ Exp(λ_i) then min{X₁, ..., X_n} ~ Exp(λ₁+ ... +λ_n).
• The Benktander Weibull distributionreduces to a truncated exponential distribution. If X ~ Exp(λ) then 1 + X ~ BenktanderWeibull(λ, 1).
• The exponential distribution is a limit of a scaled beta distribution:

{\displaystyle \lim _{n\to \infty }n\operatorname {Beta} (1,n)=\operatorname {Exp} (1).}${\displaystyle \lim _{n\to \infty }n\operatorname {Beta} (1,n)=\operatorname {Exp} (1).}$

• If X_i ~ Exp(λ) then the sum {\textstyle X_{1}+\cdots +X_{k}=\sum _{i}X_{i}\sim }${\textstyle X_{1}+\cdots +X_{k}=\sum _{i}X_{i}\sim }$ Erlang(k, λ)which is just a Gamma(k, λ⁻¹) (in (k, θ) parametrization) or Gamma(k, λ) (in (α,β) parametrization) with an integer shape parameter k.
• If X ~ Exp(1) then μ − σ log(X) ~ GEV(μ, σ, 0).
• If X ~ Exp(λ) then X ~ Gamma(1, λ⁻¹) (in (k, θ) parametrization) or Gamma(1, λ) (in (α, β) parametrization).
• If X ~ Exp(λ) and Y ~ Exp(ν) then λX − νY ~ Laplace(0, 1).
• If X, Y ~ Exp(λ) then X − Y ~ Laplace(0, λ⁻¹).
• If X ~ Laplace(μ, β⁻¹) then |X − μ| ~ Exp(β).
• If X ~ Exp(1) then (logistic distribution):

{\displaystyle \mu -\beta \log \left({\tfrac {e^{-X}}{1-e^{-X}}}\right)\sim \mathrm {Logistic} (\mu ,\beta )}$\mu-\beta\log \left(\tfrac{e^{-X}}{1-e^{-X}}\right) \sim \mathrm{Logistic}(\mu,\beta)$

• If X, Y ~ Exp(1) then (logistic distribution):

{\displaystyle \mu -\beta \log \left({\tfrac {X}{Y}}\right)\sim \mathrm {Logistic} (\mu ,\beta )}$\mu-\beta\log\left(\tfrac{X}{Y}\right) \sim \mathrm{Logistic}(\mu,\beta)$

• If X ~ Exp(λ) then ke^X ~ Pareto(k, λ).
• If X ~ Pareto(1, λ) then log(X) ~ Exp(λ).
• Exponential distribution is a special case of type 3 Pearson distribution.
• If X ~ Exp(λ) then e^−X ~ Beta(λ, 1).
• If X ~ Exp(λ) then {\displaystyle {\tfrac {e^{-X}}{k}}\sim \mathrm {PowerLaw} (k,\lambda )}$\tfrac{e^{-X}}{k} \sim \mathrm{PowerLaw}(k, \lambda)$ (power law)
• If X ~ Exp(λ) then {\displaystyle {\sqrt {X}}\sim \operatorname {Rayleigh} \left(1/{\sqrt {2\lambda }}\right).}${\displaystyle {\sqrt {X}}\sim \operatorname {Rayleigh} \left(1/{\sqrt {2\lambda }}\right).}$ (Rayleigh distribution)
• If X ~ Exp(λ) then {\displaystyle X\sim \mathrm {Weibull} ({\tfrac {1}{\lambda }},1)}$X \sim \mathrm{Weibull}(\tfrac{1}{\lambda},1)$(Weibull distribution)
• If X_i ~ U(0, 1) then

{\displaystyle \lim _{n\to \infty }n\min \left(X_{1},\ldots ,X_{n}\right)\sim {\textrm {Exp}}(1)}$\lim_{n \to \infty}n \min \left (X_1, \ldots, X_n \right ) \sim \textrm{Exp}(1)$

• If Y|X ~ Poisson(X) where X ~ Exp(λ⁻¹) then {\displaystyle Y\sim \mathrm {Geometric} ({\tfrac {1}{1+\lambda }})}$Y \sim \mathrm{Geometric}(\tfrac{1}{1+\lambda})$ (geometric distribution)
• If X ~ Exp(1) and {\displaystyle Y\sim \Gamma (\alpha ,{\tfrac {\beta }{\alpha }})}$Y \sim \Gamma(\alpha,\tfrac{\beta}{\alpha})$ then {\displaystyle {\sqrt {XY}}\sim \mathrm {K} (\alpha ,\beta )}$\sqrt{XY} \sim \mathrm{K}(\alpha,\beta)$ (K-distribution)
• The Hoyt distribution can be obtained from Exponential distribution and Arcsine distribution
• If X ~ Exp(λ) and Y ~ Erlang(n, λ) then:

{\displaystyle {\frac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}${\displaystyle {\frac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}$

• If X ~ Exp(λ) and {\displaystyle Y\sim \Gamma (n,{\tfrac {1}{\lambda }})}$Y \sim \Gamma(n,\tfrac{1}{\lambda})$ then {\displaystyle {\tfrac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}${\displaystyle {\tfrac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}$
• If X ~ SkewLogistic(θ), then log(1 + e^−X) ~ Exp(θ).
• If X ~ Exp(λ) and Y = μ − β log(Xλ) then Y ∼ Gumbel(μ, β).
• Let X ∼ Exp(λ_X) and Y ∼ Exp(λ_Y) be independent. Then {\displaystyle Z={\frac {\lambda _{X}X}{\lambda _{Y}Y}}}$Z = \frac{\lambda_X X}{\lambda_Y Y}$ has probability density function {\displaystyle f_{Z}(z)={\frac {1}{(z+1)^{2}}}}$f_Z(z) = \frac{1}{(z + 1)^2}$. This can be used to obtain a confidence interval for {\displaystyle {\frac {\lambda _{X}}{\lambda _{Y}}}}$\frac{\lambda_X}{\lambda_Y}$.
• Gamma mixture: If λ ~ Gamma(shape = k, scale = θ) and X ~ Exponential(rate = λ) then the marginal distribution of X is Lomax(shape = k, scale = 1/θ)

Other related distributions:

• Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities.
• Hypoexponential distribution – the distribution of a general sum of exponential random variables.
• exGaussian distribution – the sum of an exponential distribution and a normal distribution.

Probability density functionEdit

The probability density function (pdf) of an exponential distribution is

{\displaystyle f(x;\lambda )={\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}$

Alternatively, this can be defined using the right-continuous Heaviside step function, H(x) where H(0) = 1:

{\displaystyle f(x;\lambda )=\lambda e^{-\lambda x}H(x)}${\displaystyle f(x;\lambda )=\lambda e^{-\lambda x}H(x)}$

Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ Exp(λ).

The exponential distribution exhibits infinite divisibility.

Cumulative distribution functionEdit

The cumulative distribution function is given by

{\displaystyle F(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}$F(x;\lambda) = \begin{cases} 1-e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}$

Alternatively, this can be defined using the Heaviside step function, H(x).

{\displaystyle F(x;\lambda )=\mathrm {(} 1-e^{-\lambda x})H(x)}$F(x;\lambda) = \mathrm (1-e^{-\lambda x}) H(x)$

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