Math Is Fun Forum

saxmaestro

Member

Registered: 2007-12-09

Posts: 1

Probability Proof

Can someone help to show rho(1/2)=√(PI)  both with regular numbers and polar coordinates and that they equal each other.  (I have the proofs I just have to prove they equal each other)  rho(1/2)=(from 0 to ∞)∫[x^(-1/2)*e^-x]dx
Then let u²=x       
2udu=dx
du=(dx)/(2u);du=dx/2√(x)

2(from 0 to ∞)∫e^(-u²)du = 2[(√(PI))/2]=√PI

and then with polar coordinates

The Fonz

Member

Registered: 2007-12-09

Posts: 1

Re: Probability Proof

I'd like to know the answer to this also....any ideas?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Probability Proof

The only way I could see using polar coordinates is to do the integral from -infinity to infinity, square it, and use Fubini's theorem to convert it into a double integral, then change it into polar coordinates in the standard way.  Problem is, once you're done with this you come up with something so ridiculously messy it seems worthless to integrate.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."