52.2.9 Gamma Random Variable

The gamma distribution is a two-parameter family of probability distributions. Maxima uses the parameterization using the shape and scale for the first and second parameters of the distribution.

Function: pdf_gamma (x,a,b) ¶

Returns the value at x of the density function of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The shape parameter is \(a\), and the scale parameter is \(b\).

The pdf is

\[f(x; a, b) = {x^{a-1}e^{-x/b}\over b^a \Gamma(a)} \]

Categories: Package distrib ·

Function: cdf_gamma (x,a,b) ¶

Returns the value at x of the distribution function of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\).

The cdf is

\[F(x; a, b) = \cases{ 1-Q(a,{x\over b}) & for $x \ge 0$ \cr \cr 0 & for $x < 0$ } \]

where \(Q(a,z)\) is the gamma_incomplete_regularized function.

(%i1) load ("distrib")$

(%i2) cdf_gamma(3,5,21);
                                                 1
(%o2)        1 - gamma_incomplete_regularized(5, -)
                                                 7

(%i3) float(%);
(%o3)                 4.402663157376807e-7

Categories: Package distrib ·

Function: quantile_gamma (q,a,b) ¶

Returns the q-quantile of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\); in other words, this is the inverse of cdf_gamma. Argument q must be an element of \([0,1]\). To make use of this function, write first load("distrib").

Categories: Package distrib ·

Function: mean_gamma (a,b) ¶

Returns the mean of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The mean is

\[E[X] = ab \]

Categories: Package distrib ·

Function: var_gamma (a,b) ¶

Returns the variance of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The variance is

\[V[X] = ab^2 \]

Categories: Package distrib ·

Function: std_gamma (a,b) ¶

Returns the standard deviation of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The standard deviation is

\[D[X] = b\sqrt{a} \]

Categories: Package distrib ·

Function: skewness_gamma (a,b) ¶

Returns the skewness coefficient of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The skewness coefficient is

\[SK[X] = {2\over \sqrt{a}} \]

Categories: Package distrib ·

Function: kurtosis_gamma (a,b) ¶

Returns the kurtosis coefficient of a \(\Gamma\left(a,b\right)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The kurtosis coefficient is

\[KU[X] = {6\over a} \]

Categories: Package distrib ·

Function: random_gamma (a,b)
    random_gamma (a,b,n) ¶

Returns a \(\Gamma\left(a,b\right)\) random variate, with \(a,b>0\). Calling random_gamma with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is a combination of two procedures, depending on the value of parameter a:

For \(a>=1\), Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.

For \(0<a<1\), Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, , poisson and binomial cdf_tributions. Computing, 12, 223-246.

To make use of this function, write first load("distrib").

Categories: Package distrib · Random numbers ·