MATH 375 -- Probability and Statistics 1

Some gamma probability density functions

October 28, 2005

A continuous random variable is said to have a

gamma distribution with parameters   alpha , beta   if its

pdf has the form:  f( y ) =   y^(alpha-1)*exp(-y/beta)/(GAMMA(alpha)*beta^alpha)    if   y  > 0

                                                 0             if   y <= 0

Here are some Gamma pdf graphs.  Note that the maximum

shifts to the right and the peak value decreases as   alpha   increases.  

In the special case   alpha = 1 , the pdf is actually always decreasing for   y  > 0.

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

f := proc (y, alpha, beta) options operator, arrow; y^(alpha-1)*exp(-y/beta)/GAMMA(alpha)/(beta^alpha) end proc

beta = alpha  = 1 :

>    plot(f(y,1,1),y=0..10);

[Maple Plot]

beta = 1 , alpha = 2  :

>    plot(f(y,2,1),y=0..10);

[Maple Plot]

beta = 1 , alpha = 3  :

>    plot(f(y,3),y=0..10);

[Maple Plot]

beta = 1 , alpha = 5  :

>    plot(f(y,5),y=0..20);

[Maple Plot]

>