2. Density and Measure for Random Geometric Elements
Herbert Solomon · Society for Industrial and Applied Mathematics eBooks · 1978
In our Buffon discussion we have referred to the random positioning of a line segment in the plane and in a brief way to the similar situation for a line in the plane. For the solutions we have developed for probabilities or expectations of events, we have employed dp dθ as the appropriate density and ∫A dp dθ as the measure over some region A where p and θ are the coordinates of the line in normal form. The density dp dθ and measure based on it for events regarding lines in the plane will lead to probability and expectation statements that are invariant under the group of rigid motions in the plane, that is, translation and rotation. For some subsequent problems on lines and line segments it will be found that invariance under rotation is too demanding and we will relax this condition.