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 $\mathrm{\Gamma }(a,b)$ 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

$$f(x;a,b)=\frac{x^{a-1}{e}^{-x/b}}{b^a\mathrm{\Gamma }(a)}$$

Categories: Package distrib ·

Function: cdf_gamma (x,a,b) ¶

Returns the value at x of the distribution function of a $\mathrm{\Gamma }(a,b)$ random variable, with $a,b>0$.

The cdf is

$$F(x;a,b)=\{\begin{array}{ll}1-Q(a,\frac{x}{b})& \text{for }x\ge 0\\ \\ 0& \text{for }x<0\end{array}$$

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 $\mathrm{\Gamma }(a,b)$ 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 $\mathrm{\Gamma }(a,b)$ 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 $\mathrm{\Gamma }(a,b)$ random variable, with $a,b>0$. To make use of this function, write first load("distrib").

The variance is

$$V[X]=a{b}^2$$

Categories: Package distrib ·

Function: std_gamma (a,b) ¶

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

The skewness coefficient is

$$SK[X]=\frac{2}{\sqrt{a}}$$

Categories: Package distrib ·

Function: kurtosis_gamma (a,b) ¶

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

The kurtosis coefficient is

$$KU[X]=\frac{6}{a}$$

Categories: Package distrib ·

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

Returns a $\mathrm{\Gamma }(a,b)$ 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 ·