52.2.10 Beta Random Variable

The beta distribution is a family of distributions defined over \([0,1]\) parameterized by two positive shape parameters \(a\), and \(b\).

Function: pdf_beta (x,a,b) ¶

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

The pdf is

\[f(x; a, b) = \cases{ \displaystyle{x^{a-1}(1-x)^{b-1} \over B(a,b)} & for $0 \le x \le 1$ \cr \cr 0 & otherwise } \]

Categories: Package distrib ·

Function: cdf_beta (x,a,b) ¶

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

The cdf is

\[F(x; a, b) = \cases{ 0 & $x < 0$ \cr I_x(a,b) & $0 \le x \le 1$ \cr 1 & $x > 1$ } \]

(%i1) load ("distrib")$

(%i2) cdf_beta(1/3,15,2);
                               11
(%o2)                       --------
                            14348907

(%i3) float(%);
(%o3)                 7.666089131388195e-7

Categories: Package distrib ·

Function: quantile_beta (q,a,b) ¶

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

Categories: Package distrib ·

Function: mean_beta (a,b) ¶

Returns the mean of a \({\it Beta}(a,b)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The mean is

\[E[X] = {a\over a+b} \]

Categories: Package distrib ·

Function: var_beta (a,b) ¶

Returns the variance of a \({\it Beta}(a,b)\) random variable, with \(a,b>0\). To make use of this function, write first load("distrib").

The variance is

\[V[X] = {ab \over (a+b)^2(a+b+1)} \]

Categories: Package distrib ·

Function: std_beta (a,b) ¶

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

The standard deviation is

\[D[X] = {1\over a+b}\sqrt{ab\over a+b+1} \]

Categories: Package distrib ·

Function: skewness_beta (a,b) ¶

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

The skewness coefficient is

\[SK[X] = {2(b-a)\sqrt{a+b+1} \over (a+b+2)\sqrt{ab}} \]

Categories: Package distrib ·

Function: kurtosis_beta (a,b) ¶

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

The kurtosis coefficient is

\[KU[X] = {3(a+b+1)\left(2(a+b)^2+ab(a+b-6)\right) \over ab(a+b+2)(a+b+3)} - 3 \]

Categories: Package distrib ·

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

Returns a \({\it Beta}(a,b)\) random variate, with \(a,b>0\). Calling random_beta with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322

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

Categories: Package distrib · Random numbers ·