Difference between gamma process and gamma distribution
Answers
Answer:
gamma process is a random process with independent gamma distributed increments. Often written as {\displaystyle \Gamma (t;\gamma ,\lambda )} \Gamma (t;\gamma ,\lambda ), it is a pure-jump increasing Lévy process with intensity measure {\displaystyle \nu (x)=\gamma x^{-1}\exp(-\lambda x),} {\displaystyle \nu (x)=\gamma x^{-1}\exp(-\lambda x),} for positive {\displaystyle x} x. Thus jumps whose size lies in the interval {\displaystyle [x,x+dx)} {\displaystyle [x,x+dx)} occur as a Poisson process with intensity {\displaystyle \nu (x)dx.} \nu (x)dx. The parameter {\displaystyle \gamma } \gamma controls the rate of jump arrivals and the scaling parameter {\displaystyle \lambda } \lambda inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0.
The gamma process is sometimes also parameterised in terms of the mean ( {\displaystyle \mu } \mu ) and variance ( {\displaystyle v} v) of the increase per unit time, which is equivalent to {\displaystyle \gamma =\mu ^{2}/v} \gamma =\mu ^{2}/v and {\displaystyle \lambda =\mu /v} \lambda =\mu /v.
Explanation:
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:
With a shape parameter k and a scale parameter θ.
With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
With a shape parameter k and a mean parameter μ = kθ = α/β.
Gamma
Probability density function
Probability density plots of gamma distributions
Cumulative distribution function
Cumulative distribution plots of gamma distributions
Parameters
k > 0 shape
θ > 0 scale
α > 0 shape
β > 0 rate
Support
{\displaystyle x\in (0,\infty )} x \in (0, \infty)
{\displaystyle x\in (0,\infty )} x \in (0, \infty)
{\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}} {\displaystyle {\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}}
{\displaystyle {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}} {\displaystyle {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}
CDF
{\displaystyle {\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)} {\displaystyle {\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}
{\displaystyle {\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)} {\displaystyle {\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)}
Mean
{\displaystyle k\theta } {\displaystyle k\theta }
{\displaystyle {\frac {\alpha }{\beta }}} {\frac {\alpha }{\beta }}
Median
No simple closed form
No simple closed form
Mode
{\displaystyle (k-1)\theta {\text{ for }}k\geq 1} {\displaystyle (k-1)\theta {\text{ for }}k\geq 1}
{\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1} {\displaystyle {\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1}
Variance
{\displaystyle k\theta ^{2}} {\displaystyle k\theta ^{2}}
{\displaystyle {\frac {\alpha }{\beta ^{2}}}} {\displaystyle {\frac {\alpha }{\beta ^{2}}}}
Skewness
{\displaystyle {\frac {2}{\sqrt {k}}}} {\displaystyle {\frac {2}{\sqrt {k}}}}
{\displaystyle {\frac {2}{\sqrt {\alpha }}}} {\displaystyle {\frac {2}{\sqrt {\alpha }}}}
Ex. kurtosis
{\displaystyle {\frac {6}{k}}} {\displaystyle {\frac {6}{k}}}
{\displaystyle {\frac {6}{\alpha }}} {\displaystyle {\frac {6}{\alpha }}}
Entropy
{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}} {\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}
{\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}} {\displaystyle {\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}}
MGF
{\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}} {\displaystyle (1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}}
{\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta } {\displaystyle \left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta }
CF
{\displaystyle (1-\theta it)^{-k}} {\displaystyle (1-\theta it)^{-k}}
{\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }} {\displaystyle \left(1-{\frac {it}{\beta }}\right)^{-\alpha }}
In each of these three forms, both parameters are positive real numbers.
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and with respect to a 1/x base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).[1]
Answer:
A gamma process is a random process with independent gamma distributed increments
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are three different parametrizations in common use:
With a shape parameter k and a scale parameter θ.
With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.
With a shape parameter k and a mean parameter μ = kθ = α/β.
In each of these three forms, both parameters are positive real numbers.
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and with respect to a 1/x base measure) for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).[1]