Geodesic
On Random Variables which have the same Distribution as their Reciprocals
Canadian mathematical bulletin, Tome 8 (1965) no. 6, pp. 819-824

Voir la notice de l'article provenant de la source Cambridge University Press

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property. 
Seshadri, V. On Random Variables which have the same Distribution as their Reciprocals. Canadian mathematical bulletin, Tome 8 (1965) no. 6, pp. 819-824. doi: 10.4153/CMB-1965-062-7
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