This is just a nasty calc problem.
You want to integrate the density of 1, over the region
but only in the unit cube.
The best way to start is to find the boundary, when
and then try to make sense of your region, or the complementary region.
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.
So the required probability is ~=0.255 or 25.5%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 >
There's actually another route you can go with this. You can make use of the following fact:
If X ~ uniform[0, 1] then -log(X) ~ exponential(1).
After an annoying amount of calculus and some standard results from distribution theory, the answer I get is:
Now, -2log(y) ~ exponential(2) [I'm going to use the scale parametrizations of the exponential and gamma], and because x and z are independent, -log(x) - log(z) is the sum of two independent exponential(1) random variables, and so is distributed gamma(2, 1). So, if A ~ exponential(2) and B ~ gamma(2, 1) with A independent of B you want to calculate
It should be obvious now how to proceed. If you don't know from class the result about -log(x) giving an exponential, it is really very easy to show using the so-called distribution function method:
which is the cdf of an exponential random variable.