Inverted Distributions

Abstract

By the inverted distribution of a random variable X we mean the distribution of its reciprocal 1/X. Every distribution G that assigns probability 0 to the value 0 is the inverted distribution of some random variable, namely of X = 1/Y, where Y is distributed according to G. Thus it is impossible to assert any special properties of inverted distributions in general. As we show in Section 2, however, very weak assumptions concerning the distribution of X permit quite strong conclusions concerning G, such as Cauchy tails and multiple modes. Sections 3 and 4 discuss certain approximations and limit results concerning the inverted distributions considered in Section 2.

Key Words:

• Bimodality
• Cauchy tails
• Limit distributions
• Multimodality
• Reciprocals