Sample Probability Calculations

We can calculate probabilities for the Beta( ) distribution using either the probability density function (PDF) or the cumulative distribution function (CDF). We will denote the PDF by f and the CDF by F . The graph of the CDF for the Beta(3,4) distribution appears below, but you are encouraged to plot it for different values of and .

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

> F:=(alpha,beta,x)->int(f(alpha,beta,t),t=0..x);

> alpha:=3; beta:=4;

> plot(F(alpha,beta,x),x=0..1, title="Cumulative Distribution Function for Beta(3,4)");

> a:=plot(f(alpha,beta,x),x=0..1):

> b:=plot(f(alpha,beta,x),x=0..0.4,color=yellow,filled=true):

> display([a,b]);

Find the value of the yellow shaded area, the area under the Beta(3,4) density curve to the left of 0.4

> evalf(F(alpha,beta,0.4));

The value of this area can also be found by using the integration command.

> evalf(int(f(alpha,beta,x),x=0..0.4));

> a:=plot(f(alpha,beta,x),x=0..1):

> b:=plot(f(alpha,beta,x),x=0.4..0.6,color=yellow,filled=true):

> display([a,b]);

Find the value of the yellow shaded area, the area under the Beta(3,4) density curve between 0.4 and 0.6.

Using the CDF, we can easily find the shaded area by finding the area to the left of 0.6 and then subtracting the area to the left of 0.4.

> area:=evalf(F(alpha,beta,0.6)-F(alpha,beta,0.4));

Or, using the integration command, we can simply integrate the Beta(3,4) PDF from 0.4 to 0.6.

> evalf(int(f(alpha,beta,x),x=0.4..0.6));

> a:=plot(f(alpha,beta,x),x=0..1,labels=[" "," "]):

> b:=plot(f(alpha,beta,x),x=0.7..1,color=yellow,filled=true,labels=[" "," "]):

> c:=textplot([0.7,-.04,"x"],color=blue):

> d:=textplot([0.75,0.2,".070"],color=black):

> display([a,b,c,d]);

Find the value of x, which has .070 area to the right of it. The 3rd argument of the function call (0..1) tells Maple to look for a solution between 0 and 1.

> fsolve(F(alpha,beta,x)=.93,x,0..1);

>