Statistics for the Number of Fiber Ends in a Segment of a Random Assembly of Aligned Fibers

A plausible model is used to derive formulas for the expected number of fibers with zero, one, or two ends in a given length segment of a random assembly of aligned fibers. These results are then combined to form the expectation of the number of ends and various related quantities. By making a stronger assumption, an expression for the variance of the number of ends is found. A numerical illustration is used to show how the variance depends on the length of the segment involved. Various definitions for the proportion of ends are given and discussed. A brief discussion of the relevance of these results to the study of the mechanics of staple fiber yarns is included.