Sample Probability Calculations

We can calculate probabilities for the F( ) distribution using either the probability distribution function (PDF) or the cumulative distribution function (CDF). We will denote the PDF by f and the CDF by F . The CDF for the F(7,10) appears below, but you are encouraged to try other values of and .

> restart: with(plots):

> nu[1]:=7: nu[2]:=10:

> plot(FCDF(nu[1],nu[2],x),x=0..4, title="CDF for F(7,10)");

> f:=x->FPDF(nu[1],nu[2],x);

> F:=x->FCDF(nu[1],nu[2],x);

> a:=plot(f(x),x=0..10):

> b:=plot(f(x),x=0..2,color=yellow,filled=true):

> display([a,b]);

Find the value of the yellow shaded area, the area under the F(7,10) density curve to the left of 2.

> evalf(F(2));

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

> evalf(int(f(x),x=0..2));

> a:=plot(f(x),x=0..10):

> b:=plot(f(x),x=1..4,color=yellow,filled=true):

> display([a,b]);

Find the value of the yellow shaded area, the area under the F(7,10) density curve between 1 and 4.

Using the CDF, we find the area to the left of 4 and then subtract the area to the left of 1.

> area1:=F(4);

> area2:=F(1);

> area_between:=area1-area2;

Using the integration command, we can also integrate the FPDF from 1 to 4.

> evalf(int(f(x),x=1..4));

> a:=plot(f(x),x=0..10,labels=[" "," "]):

> b:=plot(f(x),x=1.5..10,color=yellow,filled=true,labels=[" "," "]):

> c:=textplot([1.5,-.04,"x"],color=blue):

> d:=textplot([2,0.08,".25"],color=black):

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

Find the value of x . x is known as the third quartile for the F(7, 10) distribution.

> fsolve(F(x)=.75,x);

>