I'm reading an old document and I came across a notation I'm unsure of,

assumed to be statistically and independent and uniformly distributed, viz,

$p(\phi)\mathrm{d}\phi = \frac{\mathrm{d}\phi}{2\pi}$


follows a Marshall-Palmer distribution

$N(D)\mathrm{d}D = \mathrm{e}^{-\Lambda D}\mathrm{d}(D\Lambda)$

E.g. the first equation states a uniform distribution given by $\frac{1}{2\pi}$. However, I don't understand the $\mathrm{d}\phi$ on both sides of the equation (I assume It's not a differential and if it is then what for?). The same goes for the second equation. Also, I assume that $p(\phi)$ is the notation for "the probability of a certain $\phi$, but in the second distribution, it's noted $N$? Would anyone give me an explanation of how it should be interpreted?


It would help if you could show the relevant context in the document you are reading. However, even without this, it is pretty clear that the $d \phi$ term is a differential, and presumably $\phi$ is the angle-of-rotation for some measurement so that $\phi \sim \text{U}(0, 2 \pi)$.

In Marshall and Palmer (1948) they examine data on raindrop size on dyed filter paper. They take $D$ to be the spherical diameter of raindrops and they find that the empirical data is consistent with an exponential distribution $D \sim \text{Exp}(\Lambda)$ with rate parameter $\Lambda > 0$. The density function for this random variable is $N(D) = \Lambda \exp (- \Lambda D)$. Multiplying both sides by the differential $dD$ and recognising that $\Lambda dD = d(D \Lambda)$ you then have:

$$N(D) dD = \exp (- \Lambda D) d(D \Lambda).$$

It is misleading to refer to this as a "Marshall-Palmer distribution"; it is really just an exponential distribution. Marshall and Palmer use some notation that would be considered non-standard in statistical work today, but they are essentially doing an analysis comparing raindrop size to an exponential distribution.

Marshall, J.S. and Palmer, W.M. (1948) The distribution of raindrops with size. Journal of Meteorology 5, pp. 165-166.


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