Gaussian Ratio Distribution
When X and Y are independent and have a Gaussian distribution with zero mean the form of their ratio distribution is fairly simple: It is a Cauchy distribution. However, when the two distributions have non-zero means then the form for the distribution of the ratio is much more complicated. In 1969 David Hinkley found a form for this distribution. In the absence of correlation (cor(X,Y) = 0), the probability density function of the two normal variable X = N(μX, σX2) and Y = N(μY, σY2) ratio Z = X/Y is given by the following expression:
where
The above expression becomes even more complicated if the variables X and Y are correlated. It can also be shown that p(z) is a standard Cauchy distribution if μX = μY = 0, and σX = σY = 1. In such case b(z) = 0, and
If, or the more general Cauchy distribution is obtained
where ρ is the correlation coefficient between X and Y and
The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.
Read more about this topic: Ratio Distribution
Famous quotes containing the words ratio and/or distribution:
“Official dignity tends to increase in inverse ratio to the importance of the country in which the office is held.”
—Aldous Huxley (1894–1963)
“My topic for Army reunions ... this summer: How to prepare for war in time of peace. Not by fortifications, by navies, or by standing armies. But by policies which will add to the happiness and the comfort of all our people and which will tend to the distribution of intelligence [and] wealth equally among all. Our strength is a contented and intelligent community.”
—Rutherford Birchard Hayes (1822–1893)