Suppose I have a measurement of the form µ±σ that represents a probability density function (PDF) with mean µ and confidence interval σ - basically how we're used to seeing measurements in papers. How does the arithmetic work for them? If I wanted to add or multiply two PDFs, what exactly would I do? This would be rather helpful for, for example, calculating the density of a transiting planet with reported mass and radius values.

I found a document that offered a way to multiply a univariate Guassian distribution here. I'm still toying with it. From what I've gathered so far from a programme I wrote to implement this, squaring a PDF should result in the confidence interval σ decreasing. For a sanity check, I went to a couple hot Jupiter discovery papers that report the impact parameter b as well as b^2. HAT-P-55 b's paper seems to imply that squaring a PDF shrinks its confidence interval. but the paper for HAT-P-56 b shows the confidence interval increasing when b is squared.

What is going on here? How does arithmetic work? Help

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