Originally Posted by

**dely84** Hi,

Let ksi, eta and dzeta to be iid rv ~ Uniform(0,1). What is the probability that the equation: ksi*x^2 + eta*x + dzeta = 0, to have a real solution i.e. D>=0. I got that I need to specify the limits of the tripple integral I have to solve for finding this probability... The point is that I don't know how to solve: (eta^2)/4>=ksi*dzeta. The outer integral would be from 0 to 1, but what about the inner two?

Some help will be much appreciated. Thanks.

This is quite a nice problem for demonstrating hit-and-miss Monte-Carlo. We generate a large sample of points in the unit cube count number of cases where discriminant is positive and divide by the sample size.

Code:

>NN=10000000; ..sample size
>
>xx=random(3,NN); ..sample of size NN of coefficients
>
>pp=sum(xx(1,: )^2>4*xx(2,: )*xx(3,: ))/NN ..count the positive disc and divide by sample size
0.254644
>
>
>SE=sqrt((NN*pp*(1-pp)))/NN ..approx standard error of the estimate
0.000137768
>

So the required probability is ~=0.255 or 25.5%

CB