Thread: three points question

This is just for fun, I found it on the internet but there was no 'definitive' solution.

"Three points and lie on an interval [0;1]. What is the probability that "

The suggestions there was to find probability distribution of and . In addtion, I want to apply uniform continuous distribution for each - would it be a reasonable assumption (since I am not given the distribution of these points, just the interval)?

Yeah, you'll want to assume a uniform distribution. It seems like the typical thing to do.

I got 1/3, by integrating over the appropriate region of R^3.

I was hoping I would be able to solve it myself, but I cannot. The only thing I knew how to do was to find distribution function of (x2+x3) (convolution integral), and I got two different intervals for that. How can I move from there to a function of x1-(x2+x3)?

Whoops, did it again and found a small error. I think it is 1/6.



1/3 is what you get if you incorrectly integrate up from 0 to x_2 + x_3; this should give 1/6.

If you want to do it the hard way, you should find that x_2 + x_3 has the triangle distribution on (0, 2). Then, x_1 and x_2 + x_3 are independent, and you would integrate over the appropriate area of R^2.