I'm reading a paper [1] which uses chord lengths of arbitrary 2D shapes, and they state (just before eqn 8) that
P_L(r) = (1/<l>) * Int( (l-r)*g(l), l=r..infinity)
where l is the chord length, <l> is the number average chord length, and g(l) is the probability distribution function of the chord lengths.
What is P_L(r)? What is its physical interpretation? It is stated as if this equation is an obvious thing to know.
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Some things to note: 0<l,r<infinity, and P_L is not (strictly) a probability distribution, as the paper also states that <l^2>/<l>=2*int(P_L(r), r=0..infinity)
In the paper, g(l) is the gamma distribution
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[1] Ruland, W., and B. Smarsly. 2002. "X-Ray Scattering of Non-Graphitic Carbon: An Improved Method of Evaluation." Journal of Applied Crystallography 35 (5): 624-633.
http://journals.iucr.org/j/issues/2002/05/00/pe0082/pe0082.pdf