An instrument is used to measure very small

Chapter , Problem 21

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An instrument is used to measure very small concentrations, X, of a certain chemical in soil samples. Suppose that the values of X in those soils in which the chemical is present is modeled as a random variable with density function f(x). The assay of a soil reports a concentration only if the chemical is first determined to be present. At very low concentrations, however, the chemical may fail to be detected even if it is present. This phenomenon is modeled by assuming that if the concentration is x, the chemical is detected with probability R(x). Let Y denote the concentration of a chemical in a soil in which it has been determined to be present. Show that the density function of Y is

\(g(y)=\frac{R(y) f(y)}{\int_{0}^{\infty} R(x) f(x) d x}\)