Question

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. 1. Let \( X \) be the distance of the red point to its neighbor in the counter-clockwise direction. Find the PDF of \( X \). 2. Let \( Y \) be the minimum distance of the red point to its nearest neighbor (either in the clockwise or counter-clockwise direction, whichever is smaller). Find the PDF of Y. 3. Let \( \mathrm{Z} \) be the minimum distance between any two of the points. Find the PDF of \( \mathrm{Z} \). Hint: You can represent the event \( \{z>z \) ) in terms of two random variables that are uniformly distributed on a unit square. To find \( \mathrm{P}(\mathrm{Z}>\mathrm{z}) \), determine the area that corresponds to \( \{\mathrm{Z}>\mathrm{z}) \) graphically.

          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.
1. Let \( X \) be the distance of the red point to its neighbor in the counter-clockwise direction. Find the PDF of \( X \).
2. Let \( Y \) be the minimum distance of the red point to its nearest neighbor (either in the clockwise or counter-clockwise direction, whichever is smaller). Find the PDF of Y.
3. Let \( \mathrm{Z} \) be the minimum distance between any two of the points. Find the PDF of \( \mathrm{Z} \).
Hint: You can represent the event \( \{z>z \) ) in terms of two random variables that are uniformly distributed on a unit square. To find \( \mathrm{P}(\mathrm{Z}>\mathrm{z}) \), determine the area that corresponds to \( \{\mathrm{Z}>\mathrm{z}) \) graphically.

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.
1. Let X be the distance of the red point to its neighbor in the counter-clockwise direction. Find the PDF of X.
2. Let Y be the minimum distance of the red point to its nearest neighbor (either in the clockwise or counter-clockwise direction, whichever is smaller). Find the PDF of Y.
3. 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.

Added by Robert L.

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