Joint Probability Distribution and the Minimum of a Set of Normalized Random Variables

Suppose that n types of components M1, M2… Mn are combined to form and integrated object I and suppose that y units of the integrated object are required to be formed. Assuming that not all components can be used in forming the integrated objects, let qj be the percentage of usable components of the jth type, a random variable having a probability density function fj(qj). Let wj be the normalized random variable obtained from qj by wj = qj/μj, where μj is the expected value of qj. Consider the random variable W=Min{wj, 1 ≤ j ≤ n}. This paper describes the joint probability distribution of the set of the normalized random variables and determines the probability distribution of the minimum W of this set. The expected value of W is key to determining the number of components needed to form the y integrated objects. A special case is presented where the percentages of usable components are uniformly distributed. The problem is applied to a production model.