Assume three points, one of which is red, are uniformly and independently distributed over the circumference of a circular track, which has a total length of 1. Equivalently, you can interpret the circle as a unit interval with its two ends tied together, and all distances of points are measured in terms of the length on the track. The Figure below gives a simple illustration.

Let Z be the minimum distance between any two of the points. Find the PDF of Z.

Hint: You can represent the event {Z > z} in terms of two random variables that are uniformly distributed on a unit square. To nd P(Z > z), determine the area that corresponds to {Z > z} graphically

Assume three points, one of which is red, are uniformly and independently distributed over the circum- ference of a circular track, which has a total length of 1. Equivalently, you can interpret the circle as a unit interval with its two ends tied together, and all distances of points are measured in terms of the length on the track. The Figure below gives a simple illustration. X Let Z be the minimum distance between any two of the points. Find the PDF of Z. Hint: You can represent the event (Z > z} in terms of two random variables that are uniformly distributed on a unit square. To find P(Z > z), determine the area that corresponds to {Z > z} graphically