# Estimation of BOD and DO Probability Distribution

*by*William J. Padgett, Prof. of Statistics and Mathematics; Dept. of Mathematics, Computer Sci., and Statistics, Univ. of South Carolina, Columbia, S.C.,

A. N.V. Rao, Assoc. Prof. of Mathematics; Dept. of Mathematics, Univ. of South Florida, Tampa, Fla.,

**Serial Information**:

*Journal of the Environmental Engineering Division*, 1979, Vol. 105, Issue 3, Pg. 525-533

**Document Type:**Journal Paper

**Abstract:**

Two parameters generally accepted as the primary indicators of the state of organic pollution of a stream are the biochemical oxygen demand (5-day BOD) and the dissolved oxygen (DO). A random differential equation is presented whose solution is a stochastic process representing the BOD and DO at position t downstream from the source of pollution. Using the solution of the random differential equation and bivariate probability density estimation techniques an estimate of the probability distribution of BOD and DO at position t_{1} along a stretch of stream is derived based on n observed values of BOD and DO at that point. The estimated distribution is used to obtain the probability distribution of BOD and DO at other points t along the stretch and to obtain probability bounds on the BOD and DO at t. In particular, the probability of DO below a minimum specified level can be computed in order to control the pollution input. A numerical example is given for a typical stream.

**Subject Headings:**Probability | Probability distribution | Rivers and streams | Oxygen demand | Dissolved oxygen | Differential equations | Stochastic processes | Water pollution

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