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I have a neophyte question regarding the formulation of the Bertrand's paradox. This is related to the definition of what we call a uniform distribution. If we consider the 3 point of views of “random endpoints” method, “random radius” method and “random midpoint” method, is the randomness (supposed to be uniform in each case) equivalent between the three methods? Indeed a simple case considering a random variable X to be uniform. The new variable Y defined as Y=X^2 is not uniformly distributed because it is the argument X which is uniform.

My question is: are the three methods equivalent in term of randomness ? The uniform distribution in 1st method may induce perhaps a bias in the equivalent randomness in the 2 other methods. I don't know how to demonstrate it mathematically. If this is true, each method supposes the cord being uniformly distributed but indeed, they are not equivalent. Should it be a a ill-problem ? because we need to define what we call "uniformly distributed" and w.r.t. what Space subset domain. Thank you for considering my question.

Alain Bensoussan
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  • Define the pdf in each case, These would be equal. – herb steinberg May 17 '22 at 21:51
  • Method Random endpoints: Pick two random points on the circumference of the circle. In this case the Space subset is the circumference of the circle.

    Method Random radius: Pick a random radius of the circle and then a random point on the radius. Construct the unique chord perpendicular to the point. The subset is the area of the circle including the circumference. But 2 randomness.

    Method Random midpoint: Pick a point inside the circle at random and take this to be the midpoint of the chord perpendicular to the radius containing it. The subset same as method 2 but 1 randomness.

     – Alain Bensoussan May 17 '22 at 21:57

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Jaynes gives a fairly good treatment of the paradox with an approach based on his principle of transformation groups. I think you'll find his take satisfying, given the last few sentences of your question.

This skeptical reading/re-analysis of Jaynes' argument is also quite good.

Above all: if confusion threatens when trying to rigorously interpret statements of probability, always remember that we don't have to.

user3716267
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