A Small Problemette related to Cosmological non-Gaussianity
Writing yesterday’s post I remembered doing a calculation a while ago which I filed away and never used again. Now that it has come back to my mind I thought I’d try it out on my readers (Sid and Doris Bonkers). I think the answer might be quite well known, as it is in a closed form, but it might be worth a shot if you’re bored.
The variable 
where 
Answers on a postcard through the comments box please..
In the Dark 
April 8, 2013 at 12:34 pm
Depends which branch(es) of the square root is/are physical.
April 8, 2013 at 12:34 pm
Is it possible to have a large problemette?
M.
April 8, 2013 at 3:27 pm
I can write down a very messy closed-form answer:
f(y) = g(x1) / |y'(x1)| + g(x2) / |y'(x2)|
where where x1,x2 are the roots of the quadratic and g is the Gaussian pdf of x. The above expression is valid for y > ymin, where ymin is the smallest value for which the quadratic has real roots; f(y) is zero for values below this.
Is there a nicer way to write it? Something that provides more insight?
April 8, 2013 at 3:37 pm
There is an explicit expression in terms of special functions…
April 9, 2013 at 6:46 pm
Is it better (simpler, clearer, more useful) than an explicit expression in terms of elementary functions (i.e., the expression I gave above, after making the described substitutions)?
April 8, 2013 at 4:49 pm
Interesting how the pdf for y reverts to Gaussian as alpha tends to zero…
April 10, 2013 at 8:14 am
The nicest way I know to change variables in probability distributions, say from x to y=f(x), is to write the distribution for the new variable, P(y), as the marginal of the joint distribution, \int dx P(x,y) , use the product rule to decompose the joint distribution as P(y|x)P(x), and write the conditional distribution as
P(y|x) = \delta (y – f(x)).
Then you can substitute in any handy representation of the delta-function, such as its spectral representation
\delta (z) = 1/(2\pi) \int dk \exp (-ikz)
and interchange the order of integration over x and k in the resulting expression for P(y). In the present problem, you get the answer that Ted gave by performing the x-integration and then using contour integration and the residue theorem to do the k-integration.
But I too would like to see how and why Peter advocates a special-function expression…
April 10, 2013 at 2:42 pm
I don’t really, but it’s basically a form of the noncentral chi-squared distribution of order unity so if you’re old-fashioned you can use the tables that exist for that function.
April 10, 2013 at 1:48 pm
Presumably he defines the answer as C(y), where C(y) is a new special function (the Coles function). All special functions are effectively this copout: you can’t solve the problem, so you give a name to your failure. But I suppose even non-special functions share this stigma: as those who are old enough to remember log tables can testify, there isn’t a finite computation that will give you ln(x) or sin(x). But some cases are worse than others: my blood pressure tends to go off the scale every time Mathematica claims to have solved an equation, only then to serve up some variant of the bloody hypergeometric function. I wish the EU would ban it.
April 10, 2013 at 2:26 pm
How are you supposed to get your money’s worth out of Gradshteyn & Ryzhik if you don’t get to write things like confluent hypergeometric functions all over the place?
April 12, 2013 at 6:07 am
Ted’s solution involving elementary functions can be obtained using elementary forces, so presumably Peter’s solution involving special functions used Special Forces. I wondered if this meant the SAS, but searched in vain for an updated edition of Abramowitz and Stegun with a third author. The Coles function C(y) clearly needs two arguments after renormalising the x field to unit variance. However, little is known about C(alpha,y), except that C(2,Newcastle) must be an integer (as it’s pointless), and that C(4,Prime Minister) is purely imaginary.
April 12, 2013 at 9:05 am
There IS an update of Abramowitz and Stegun as of a year or two ago, by NIST.