Mean Free Path — Question 3

Question 3

An exponential free-path model predicts survival and quantiles

In a supplied model, the probability of traveling at least distance $L$ without collision is $S(L)=e^{-L/\lambda}$, with $\lambda=80.0$ nm. Flights are statistically independent and the medium is uniform.

1. Find the probability of no collision over 200 nm and the distance exceeded by half of all free flights.

2. Find the distance over which 95.0% of trajectories have suffered at least one collision. Is the mean free path a maximum possible flight?

Question 3 - Solution

An exponential free-path model predicts survival and quantiles

(a) Method and calculation

$$S(200\,\mathrm{nm})=e^{-200/80}=e^{-2.5}\approx.0820850.$$$$S(L_{50})=.500\ \Rightarrow\ L_{50}=\lambda\ln2\approx55.4518\,\mathrm{nm}.$$

(b) Interpretation and further calculation

$$1-S(L_{95})=.950\ \Rightarrow\ L_{95}=-\lambda\ln(.0500)=80\ln20\approx239.659\,\mathrm{nm}.$$$$S(\lambda)=e^{-1}\approx.367879.$$Many flights exceed the mean, and the ideal exponential distribution has no finite upper cutoff.

Physical check

The median is smaller than the mean because the distribution has a long positive tail.