The Use of Incomplete Beta Functions for Prior Distributions in Binomial Sampling

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Abstract
Let θ be the proportion of defectives in a batch selected at random from a fixed population of batches. The paper deals with the determination of the prior distribution of θ, using prior knowledge obtained from batches examined in the past. Assuming a beta type prior distribution, the following three situations are investigated: a. Past records provide the numbers r 1, r 2, …, r N of defectives found in N samples of size n from N previously inspected batches;b. The expected fraction defective in a batch, E(θ), and the probability, P, of a batch exceeding twice the expected fraction defective are approximately known;c. (c) The probabilities of θ exceeding a certain value θ1 and of θ falling below another value θ2 are approximately known. Past records provide the numbers r 1, r 2, …, r N of defectives found in N samples of size n from N previously inspected batches; The expected fraction defective in a batch, E(θ), and the probability, P, of a batch exceeding twice the expected fraction defective are approximately known; (c) The probabilities of θ exceeding a certain value θ1 and of θ falling below another value θ2 are approximately known. Methods of estimating the parameters of the prior distribution are given, and charts are provided to facilitate their determination. The effects of errors in the parameters on the posterior distribution are discussed in relation to sample size and number of defectives found in the sample. The effects of assuming a beta distribution when in fact some other distribution would have been appropriate are found to be negligible in most practical applications.

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