Unexpected distribution of ab-cd where a,b,c,d are independent and N(0,1) distributed I have discovered an unexpected curiosity that if
a,b,c,d are independent random variables and are N(0,1) distributed, then
|ab-cd| is exponentially distributed Exp(1). It's easy to verify that numerically.
Is there a proof of that ? or does it stem from some deeper fact ?
I noticed that it when I was considering cross products of random Gaussian distributed vectors and noticed that their components have \exp(-|x|/a) distribution.
Demonstration with python
N = 1000000
x = np.random.normal(size=(N,4))
z = np.abs(x[:,0] * x[:,1] - x[:,2] * x[:,3])
plt.hist(scipy.stats.expon.cdf(z), bins=100)



There is an extensive theory on the properties of random matrices, including the distribution of their determinants.
From the answer here for example you can see that if you form the matrix $W=AA^T$, where the elements of $A$ are your $\mathcal N (0,1)$ random variables, then
$$ \det W \sim \chi^2_2 \chi^2_1 $$
namely the determinant of $W$ has the distribution of a product of two independent $\chi^2$ random variables with 2 and 1 degrees of freedom.
The distribution of the product can be derived using the equivalent distribution of a product of Gamma random variables. In the general case this is expressed in terms of a modified Bessel functions, but here it reduces to a simpler expression, and using the transformation $|\det A|=\sqrt{\det W}$ it can be worked out quite easily that indeed $|\det A|$ has an exponential distribution with mean 1.
There might be a more direct and maybe simpler way of driving this property that i'm missing at the moment - maybe someone else can find it.
