MATH 375 -- Probability and Statistics 1

Beta distributions

November 3, 2003

The beta pdf with parameters alpha, beta   has the form

     f(y) = y^(alpha-1)*(1-y)^(beta-1)/B(alpha,beta)    if  0 <= y <= 1  

and 0 otherwise, where   B(alpha,beta) = GAMMA(alpha)*GAMMA(beta)/GAMMA(alpha+beta)  .   Here

are some examples, using the BetaPDF function from

our Maple package.

>    read "/home/fac/little/public_html/ProbStat/MaplePackage/MSP.map";

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Here is a typical shape:

>    plot([y->BetaPDF(6.7,2.3,y)],0..1,color=[red]);

[Maple Plot]

>   

Fixing alpha   and increasing   beta   shifts the location of the peak to the left, and narrows it

>    plot([y->BetaPDF(3,1,y),y->BetaPDF(3,3,y),y->BetaPDF(3,5,y),y->BetaPDF(3,7,y)],0..1,color=[red,blue,black,gray]);

[Maple Plot]

Fixing beta   and increasing alpha   shifts the location of the peak to the right, and narrows it

>    plot([y->BetaPDF(1,3,y),y->BetaPDF(3,3,y),y->BetaPDF(5,3,y),y->BetaPDF(7,3,y)],0..1,color=[red,blue,black,gray]);

[Maple Plot]

Indeed, it is easy to see that the beta pdf with parameters alpha = a, beta = b   and the

pdf with parameters alpha = b, beta = a   have graphs that are reflections of one another

across the line   y = 1/2   (do you see why?)