Late night maths

[Inquisitor]

I'm trying to integrate the Gaussian distribution between arbitrary limits, but I'm not having a lot of luck. As far as I can see I've done it right, but the answer I get is imaginary, which is obviously wrong, since it's supposed to represent a probability

Steps I'm taking:

1. Turn it into a double integral over x and y
2. Transform to polar coordinates; dxdy becomes rdrdθ and the limits become the corresponding values of r and θ for x=b, x=a (do I need to do something else with the θ limits perhaps?)
3. Evaluate the r (inner) integral (with respect to r) and bring it outside the outer integral as a coefficient, since it's constant (is this part right? I'm not quite sure)
4. Evaluate the θ integral; this just becomes θ(b) - θ(a).

Here's my working:

The question I'm doing has the values of a, b, α and β as 299, 301, 2.37e-6, and -6.4e-6 respectively. This gives me an answer of 0.513e-5 i, which is clearly wrong (it should be around 2.84e-3).

Can anyone tell me what I'm doing wrong?

Edit: I meant to post this on physicsforums.com but absent mindedly did it in the wrong tab, help still appreciated though

 

[Rebelius]

if I stick all that into maple, I just get 0.2664556054e-5


that's int(alpha*exp(beta*x^2), x = a .. b) with your values for a,b,alpha,beta.


I haven't actually done enough maths/stats recently to be much use to you though, halfway through a gap year half way through my degree.

 

[Inquisitor]

Whoops, α should be 0.00143, sorry

If you're using Maple it might help to know α and β exactly, actually:
α = sqrt(1/(156250π))
β = -1/156250

I just got Maple to evaluate it for me using those values and got 0.00160, which sounds plausible. Clearly I'm doing something wrong

 

[Rebelius]

with those numbers maple gives 1.6e-3

as I said, actually doing the maths is a bit beyond my ability just now, I'm well out of practice. Sorry.

 

[Inquisitor]

No worries, thanks anyway