Читать книгу Probability and Statistical Inference - Robert Bartoszynski - Страница 53

Imagine different devices (physical mechanisms, computer programs, etc.) built for sampling random chords. One scheme chooses a point on the circumference, and then the angle between the chord and the tangent to the circle at the point chosen (Figure 2.2). The second scheme chooses first the direction of the diameter and then the point on the diameter, at which the chord perpendicular to this diameter intersects it (Figure 2.3). From Figure 2.5, it is seen that the angle is , and therefore . Thus, , which means that equal changes of do not produce equal changes of . In fact, these changes are smaller when is small. Consequently, a device that chooses angles at random will tend to produce more intersections of the diameter that are farther from the center (i.e., more chords will be shorter).