Archive for convolution

mixtures of sums vs. sum of mixtures

Posted in Statistics with tags , , , on April 13, 2022 by xi'an

A (mildly) interesting question on X validated last nigh, namely the distribution of a sum of n iid variables distributed from a mixture of exponentials. The rather obvious answer is a mixture of (n+1) distributions, each of them corresponding to a sum of two Gamma variates (but for the extreme cases). But the more interesting component for my personal consumption is that the distribution of this sum of two Gammas with different scales writes up as a signed mixture of Gammas, which comes as an handy (if artificial) illustration for a paper we are completing with Julien Stoehr.

Gauss to Laplace transmutation interpreted

Posted in Books, Kids, Statistics, University life with tags , , , , , , on November 9, 2015 by xi'an

Following my earlier post [induced by browsing X validated], on the strange property that the product of a Normal variate by an Exponential variate is a Laplace variate, I got contacted by Peng Ding from UC Berkeley, who showed me how to derive the result by a mere algebraic transform, related with the decomposition

(X+Y)(X-Y)=X²-Y² ~ 2XY

when X,Y are iid Normal N(0,1). Peng Ding and Joseph Blitzstein have now arXived a note detailing this derivation, along with another derivation using the moment generating function. As a coincidence, I also came across another interesting representation on X validated, namely that, when X and Y are Normal N(0,1) variates with correlation ρ,

XY ~ R(cos(πU)+ρ)

with R Exponential and U Uniform (0,1). As shown by the OP of that question, it is a direct consequence of the decomposition of (X+Y)(X-Y) and of the polar or Box-Muller representation. This does not lead to a standard distribution of course, but remains a nice representation of the product of two Normals.

Gauss to Laplace transmutation!

Posted in Books, Kids, Statistics, University life with tags , , , , on October 14, 2015 by xi'an

When browsing X validated the other day [translate by procrastinating!], I came upon the strange property that the marginal distribution of a zero mean normal variate with exponential variance is a Laplace distribution. I first thought there was a mistake since we usually take an inverse Gamma on the variance parameter, not a Gamma. But then the marginal is a t distribution. The result is curious and can be expressed in a variety of ways:

– the product of a χ21 and of a χ2 is a χ22;
– the determinant of a 2×2 normal matrix is a Laplace variate;
– a difference of exponentials is Laplace…

The OP was asking for a direct proof of the result and I eventually sorted it out by a series of changes of variables, although there exists a much more elegant and general proof by Mike West, then at the University of Warwick, based on characteristic functions (or Fourier transforms). It reminded me that continuous, unimodal [at zero] and symmetric densities were necessary scale mixtures [a wee misnomer] of Gaussians. Mike proves in this paper that exponential power densities [including both the Normal and the Laplace cases] correspond to the variances having an inverse positive stable distribution with half the power. And this is a straightforward consequence of the exponential power density being proportional to the Fourier transform of a stable distribution and of a Fubini inversion. (Incidentally, the processing times of Biometrika were not that impressive at the time, with a 2-page paper submitted in Dec. 1984 published in Sept. 1987!)

This is a very nice and general derivation, but I still miss the intuition as to why it happens that way. But then, I know nothing, and even less about products of random variates!

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