Correlated gamma variables in the analysis of microbial densities in water

The probability density function (p.d.f.) of the ratio of two correlated gamma variables is derived and used to fit aquatic microbial-density data. The ratio p.d.f. is tackled by first taking the Fourier transform of a generalized Kibble–Gaver, unsymmetrical, characteristic function (c.f.) to obtain the corresponding bivariate p.d.f. of two correlated gamma variables with different shape and scale parameters. The ratio p.d.f. follows by weighted integration of the bivariate p.d.f. The derivation of the gamma bivariate and ratio p.d.f.s relies on the use of weighted Laguerre–Charlier polynomials that lead to p.d.f.s amenable to computation. The bivariate gamma p.d.f. and the ratio p.d.f. of correlated gamma variables are useful statistical tools in the analysis of skewed water-resources data. Computational examples illustrate the calculation of bivariate p.d.f.s for positive and negative correlation and the fitting of the ratio p.d.f. to correlated bacterial densities in stream water.