distribution of the ratio of jointly normal variables

Abstract

We derive the probability density of the ratio of components of the bivariate normal distribution with arbitrary parameters. The density is a product of two factors, the first is a Cauchy density, the second a very complicated function. We show that the distribution under study does not possess an expected value or other moments of higher order. Our particular interest is focused on the shape of the density. We introduce a shape parameter and show that according to its sign the densities are classified into three main groups. As an example, we derive the distribution of the ratio Z = − Bm−1 /(mBm ) for a polynomial regression of order m. For m=1, Z is the estimator for the zero of a linear regression, for m = 2 , an estimator for the abscissa of the extreme of a quadratic regression, and for m = 3 , an estimator for the abscissa of the inflection point of a cubic regression.

Limits of emission quantum yield determination

“…It then follows that the derived ratio Z = X/Y is a continuously distributed random variable with the probability density function (for more details, see the Supplementary Materials) 20 :…”

“…The second factor in the brackets is the 'deviant part', which is a strictly positive and asymptotically constant function D(z) 20 : lim z→±∞ D(z) = const( ρ, α X , α Y , θ )…”

When is Chemical Similarity Significant? The Statistical Distribution of Chemical Similarity Scores and Its Extreme Values

“…The density of the ratio of two correlated normal random variables has been studied in the literature and can be obtained analytically, although its expression is somewhat involved. − The probability density for Z = X / Y , where X ≈ 𝒩(μ X , σ X 2 ), Y ≈ 𝒩(μ Y , σ Y 2 ), and ρ = Corr( X , Y ) ≠ ±1 is given by the product of two terms: P Z ( z ) = σ X σ Y 1 − ρ 2 π ( normalσ Y 2 z 2 − 2 normalρ normalσ X normalσ Y z + normalσ X 2 ) true[ exp true( − 1 2 sup R 2 true) × newline true( 1 + R Φ ( R ) ϕ ( R ) true) true] or P Z ( z ) = σ …”

“…The density of the ratio of two correlated normal random variables has been studied in the literature and can be obtained analytically, although its expression is somewhat involved. [24][25][26][27] The probability density for Z ) X/Y, where…”

Run sum ratio chart when measurement errors are present

“…respectively. ρ denotes the correlation coefficient between X and Y. Additionally, the CVs for X and Y are given by 𝛾 𝑋 = Studies in the past have looked into the distribution of the ratio of two variables, both belonging to a normal distribution, for example, see Hayya et al 27 and Cedilnik et al 28 The distribution of 𝑍 = 𝑋 𝑌 can be approximated using the approach presented in either Hayya et al 27 or Celano and Castagliola. 4 This article approximates the cumulative distribution function (cdf) of Z as follows (Celano and Castagliola 4 ):…”