Properties of the Distribution

Calculation of Mean and Variance

To calculate the mean of the Beta distribution, we can simply integrate x *(Beta PDF), per the definition of mathematical expectation.

> restart;

> with(plots, display):

> interface( showassumed = 0 );

> assume(alpha>0); assume(beta>0);

> f:=(alpha,beta,x)->GAMMA(alpha+beta)/(GAMMA(alpha)*GAMMA(beta))*x^(alpha-1)*(1-x)^(beta-1);

> EX:=int(x*f(alpha,beta,x),x=0..1);

Calculating Var( X ), the variance of the Beta( ) distribution, we will employ the formula: Var( X ) = E( ) -

> E_X_SQ:=int((x^2)*f(alpha,beta,x),x=0..1);

> VarX:=simplify(E_X_SQ-EX^2);

>