Ratios of Random Variables

Abstract

Let ${\rm X}_1 \in {\rm N}_1 \left( {0,\sigma _1^2 } \right)$ and ${\rm X}_2 \in {\rm N}_1 \left( {0,\sigma _2^2 } \right)$ be independent Gaussian RVs. Then, the ratio of these RVs X = X₂/X₁, has the following statistical properties.

$$ p_X \left( x \right) = \frac{{\sigma _1 \sigma _2 }} {{\pi \left( {\sigma _{_1 }^2 x^2 + \sigma _2^2 } \right)}} $$

(7.1)

$$ p_X \left( x \right) = \frac{1} {2} + \frac{1} {\pi }\tan ^{ - 1} \left( {\frac{{\sigma _1 x}} {{\sigma _2 }}} \right) $$

(7.2)

$$ \Psi _X \left( \omega \right) = \exp \left( { - \frac{{\sigma _2 }} {{\sigma _1 }}\left| \omega \right|} \right) $$

(7.3)

A large number of the CDF results in this section come from Ref. 11 where they appear in their normalized (the RVs that form the ratio have unit variance) form. Once again, a large number of the PDF and statistical moment results come from Ref. 5.