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The beta distribution is a family of distributions defined over [0,1] parameterized by two positive shape parameters a, and 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 $$ f(x; a, b) = \cases{ \displaystyle{x^{a-1}(1-x)^{b-1} \over B(a,b)} & for $0 \le x \le 1$ \cr \cr 0 & otherwise } $$
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
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").
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 $$ E[X] = {a\over a+b} $$
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 $$ V[X] = {ab \over (a+b)^2(a+b+1)} $$
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 $$ D[X] = {1\over a+b}\sqrt{ab\over a+b+1} $$
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 $$ SK[X] = {2(b-a)\sqrt{a+b+1} \over (a+b+2)\sqrt{ab}} $$
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 $$ KU[X] = {3(a+b+1)\left(2(a+b)^2+ab(a+b-6)\right) \over ab(a+b+2)(a+b+3)} - 3 $$
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").