Multiple Sampling for Variables

Abstract

A multiple (sequential) sampling scheme designed to test certain hypothesis is discussed with the following assumption: $X$ is a random variable with density function $P(x)$ which is piecewise continuous and differentiable at its points of continuity. Formulae are derived for the probability of acceptance and rejection of the hypothesis and for the expected number of samples necessary for reaching a decision. These formulae are found to depend on the solution of a Fredholm Integral equation. Explicit solutions to the problem are obtained for the case when $P(x)$ is rectangular by reducing the fundamental integral equation to a set of differential-difference equations. Several examples are given.