Prob help
From PIMC++
For Normal (Gaussian) Distributions, given two independent data sets,
,
then combining the two data sets we get
,
.
The error when adding (subtracting) the two sets can be understood
by considering a Gaussian in two independent variables--a 2 dimensional
Guassian--centered at the origin.
Since the error in x and the error in y are the deviation from zero in
orthogonal directions, they combine through the Pythagorean relation.
The Normal Distribution Function, 
is the probability that a point taken from a Normal Distribution,
with variance 
and mean 
, is between 
and 
.
(Note that here we have
added the subscript u for "unscaled", while we will reserve the notation
p(x) for the "scaled", Standard Normal Distribution Function.)
The Normal
Distribution Function is rescaled to become the Standard Normal Distribution
Function through the change of variables,
.
Dropping the tildes we get the Standard Normal Distribution Function with
mean zero and x rescaled so it is now the number of standard deviations
from the mean:
.
Also note that since the integrand is normalized and an even function,
and 
.
We have defined the standard normal distribution function to allow us to
vary both limits of integration. Doing this gives
,
where 
is the error function,
.
Note that 
is the probability of finding a point within standard deviations from the mean while 
is the probability of finding a point outside standard deviations from the mean. The values for both
and can be found in tables (such as the standard Abramowitz and Stegun, Handbook of Mathematical Functions) and are often included
as functions in math programs, such as Mathematica.
(In Mathematica use Erf[x], and don't forget about the square root of 2.)
For this problem, find the difference between the means of the two data
sets and divide by their combined error; call this a. The probability
that the two data sets are from the same distribution is 
.
Equivalently, the probability that they are NOT from the same distribution is 
.
