3 ms·
Though it should be noted that "cubing" doesn't refer to just taking the cube, i.e if f(x) = e^{-x^2/2} is proportional to the standard normal distribution then
by tgb 9y ago
Though it should be noted that "cubing" doesn't refer to just taking the cube, i.e if f(x) = e^{-x^2/2} is proportional to the standard normal distribution then the cube of f(x), namely f * f * f is not the distribution you're considering here. Rather it's the "pushforward measure" [1], in this case is 1/3 y^{-2/3} e^{-y^{2/3}/2} where y=x^3 is the mass and x is the radius. (Up to constants.)
[1] https://en.wikipedia.org/wiki/Pushforward_measure https://en.wikipedia.org/wiki/Pushforward_measure
- xyzzyz 9y agoCubing does refer to just taking the cube, but the cube of random variable, and not its probability density function -- the density after cubing is indeed the pushforward.
- bonoboTP 9y agoThis is indeed one confusion that I ran into very often when first learning stats. Transforming the random variable is a very different thing than transforming the probability density function directly. The distinction between a random variable and a distribution (or density) is not made clear enough in classes I think.