Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\dots,n$, where $U_{(0)}:=0$ and $U_{(n+1)}:=1$.

It is now a textbook fact that the joint distribution of $G_0,\dots,G_n$ is the same as that of $R_1,\dots,R_{n+1}$, where $R_i:=H_i/(H_1+\dots+H_{n+1})$ and the $H_i$'s are iid standard exponential random variables.

Moran, page 93 ascribes mentioning of this fact to Fisher, and a proof of it -- without a specific reference -- to Clifford.

Thus, here is my question:

Can one give a reference to that paper by Clifford, assuming it does exist?

  • $\begingroup$ Moran cites Clifford with a footnote for an 1876 contribution reprinted in Clifford’s collected works. $\endgroup$ – Matt F. Aug 28 at 0:37
  • $\begingroup$ @MattF., you beat me to it! $\endgroup$ – kodlu Aug 28 at 0:38

In case @IosifPinelis doesn't have access to p.98 of Moran, the full reference is

Clifford, W. K., (1866): "Solution to Problem 1878", Educational Times, Jan., Reprinted in Mathematical Papers, pp. 601--607.

  • $\begingroup$ Thank you. I have now found Clifford's "Solution to Problem 1878", beginning on page archive.org/details/mathematicalpap00smitgoog/page/n707 . However, that Solution seems to be for another problem, also concerning a line "broken into $n$ random pieces". Alas, I have been unable to find the desired result in that Solution or indeed in that entire collection of Clifford's papers. Perhaps it is not accidental that Moran does not specifically reference Clifford where he (Moran) talks about the relation with exponential random variables. $\endgroup$ – Iosif Pinelis Aug 28 at 1:37
  • $\begingroup$ That's interesting, so the connection is a bit more tenuous $\endgroup$ – kodlu Aug 28 at 1:49

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