NationMaster - Encyclopedia: Inverse gamma distribution

**Inverse-gamma**
Probability density function
Cumulative distribution function
Parameters	$α > 0$ shape (real) $β > 0$ scale (real)
Support	$xin(0;infty)!$
pdf	$frac{beta^alpha}{Gamma(alpha)} x^{-alpha - 1} exp left(frac{-beta}{x}right)$
cdf	$frac{Gamma(alpha,beta/x)}{Gamma(alpha)} !$
Mean	$frac{beta}{alpha-1}!$ for $α > 1$
Median
Mode	$frac{beta}{alpha+1}!$
Variance	$frac{beta^2}{(alpha-1)^2(alpha-2)}!$ for $α > 2$
Skewness	$frac{4sqrt{alpha-2}}{alpha-3}!$ for $α > 3$
Kurtosis	$frac{30,alpha-66}{(alpha-3)(alpha-4)}!$ for $α > 4$
Entropy	$alpha!+!ln(betaGamma(alpha))!-!(1!+!alpha)psi(alpha)$
mgf	$frac{2left(-beta tright)^{!!frac{alpha}{2}}}{Gamma(alpha)}K_{alpha}left(sqrt{-4beta t}right)$
Char. func.	$frac{2left(-ibeta tright)^{!!frac{alpha}{2}}}{Gamma(alpha)}K_{alpha}left(sqrt{-4ibeta t}right)$

The inverse gamma distribution has the probability density function over the support $x > 0$ Download high resolution version (1299x966, 152 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x965, 178 KB) See the image on the commons for gnuplot source. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... Jump to: navigation, search In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... Jump to: navigation, search In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... Some mathematicians use the phrase characteristic function synonymously with indicator function. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

with shape parameter $α$ and scale parameter $β$ . In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ...

The cumulative distribution function is In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...

$F(x; alpha, beta) = frac{Gamma(alpha,beta/x)}{Gamma(alpha)} !$

where the numerator is the upper incomplete gamma function and the denominator is the gamma function. In mathematics, the gamma function is defined by a definite integral. ... The Gamma function along an interval In mathematics, the Gamma function extends the factorial function to the complex numbers. ...

Related distributions

$X sim mbox{Inv-Gamma}(alpha, beta)$ then $Y sim mbox{Inv-chi-square}(nu)$ if $alpha = frac{nu}{2}$ and $beta = frac{1}{2}$ ; $Y$ is the Inverse-chi-square distribution
$X sim mbox{Gamma}(k, theta)$ then if $Y = frac{1}{X}$ ; $X$ is the Gamma distribution

In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

Gamma distribution derivation

The pdf of the gamma distribution is

$f(x) = x^{k-1} frac{e^{-x/theta}}{theta^k , Gamma(k)}$

and define the transformation $Y = g(X) = frac{1}{X}$ then the resulting transformation is

$f_Y(y) = f_X left( g^{-1}(y) right) left| frac{d}{dy} g^{-1}(y) right|$

$= frac{1}{theta^k Gamma(k)} left( frac{1}{y} right)^{k-1} exp left( frac{-1}{theta y} right) frac{1}{y^2}$

$= frac{1}{theta^k Gamma(k)} left( frac{1}{y} right)^{k+1} exp left( frac{-1}{theta y} right)$

$= frac{1}{theta^k Gamma(k)} y^{-k-1} exp left( frac{-1}{theta y} right)$

Replacing $k$ with $α$ ; $θ - 1$ with $β$ ; and $y$ with $x$ results in the inverse-gamma pdf shown above

$f(x) = frac{beta^alpha}{Gamma(alpha)} x^{-alpha-1} exp left( frac{-beta}{x} right)$

Related distributions var ACE_AR = {Site: '756743', Size: '300250'};

Gamma distribution derivation

See also

Related distributions