3.2.1 Transformation of probability densities

Given $p(x)$ and a bijective mapping $y=f(x); \; x=f^{-1}(y)$; then

$$\vert p(y) \, dy \vert = \vert p(x)\,dx\vert$$

or

$$p(y) = p(x) \, \left\vert \frac{dx}{dy}\right\vert = p[f^{-1}(y)] \, \left\vert \frac{df^{-1}(y)}{dy} \right\vert$$

This relation holds for any kind of density.

EXAMPLE: The spectral density of black body radiation is usually written in terms of the angular frequency $\omega$:

$$I(\omega) = \frac{\hbar \omega^{3}}{\pi c^{3}} \frac{1}{e^{\hbar \omega/kT}-1}$$

If we prefer to give the spectral density in terms of the wave length $\lambda \equiv 2 \pi c / \omega$, we have

$$I(\lambda) = I[\omega(\lambda)] \, \left\vert \frac{d \omega}{d \lambda}\right... ...)^{3} \frac{1}{e^{(hc/\lambda)/kT}-1} \left( \frac{2 \pi c}{\lambda^{2}}\right)$$

EXERCISE: A powder of approximately spherical metallic grains is used for sintering. The diameters of the grains obey a normal distribution with $\langle d \rangle =2 \mu m$ and $\sigma=0.25 \mu m$. Determine the distribution of the grain volumes.