Stick breaking problem - Intuition

[dawg]

Hi everyone,
My question is regarding the classic example commonly taught with probability basics, a stick is given and we are told it is broken at some point along its length, at random. There are 2 questions i've come across for this setup, and what confuses me is that the mismatch between their answers when intuitively they should match.
Assume for simplicity the length of the stick is 1

The first: the random variable X is the position of the break along the stick. Find the expectation of X. This will obviously be at 1/2 since X~Unif(0,1)

The second: Find the expectation of the ratio of the lengths of the two pieces. Here X is the length of one of the pieces, so (using L as the unit of length so the stick is of length 1) 1-X is the length of the second piece. Now this bit confuses me, the PDF of this ratio is constant at 1/2, but it isn't uniformly distributed since the expectation isn't 1/4, instead, the integration occurs over x, not over y=x/1-x, and so the answer is something like 0.38. In both cases (1/4 or 0.38) this doesn't match the expected breaking point from the first question which would lead to two pieces of the same length or (x/1-x) = 1.

My question is, why is it that asking two seemingly equivalent questions, lead to two different answers ? I have been thinking about this a lot and cannot find an intuitive explanation for this.

 

[Steven G]

I too need to think about this. But please stop writing x/1-x = 1 as x/1-x = x-x=0. Maybe you meant x/(1-x)?

 

[dawg]

yes I meant 1/(1-x), apologies.

 

[Dr.Peterson]

Please explain what you mean by this:

the PDF of this ratio is constant at 1/2, but it isn't uniformly distributed

I have to say, though I haven't put any thought into the details, I would never expect your two questions to have related answers, though I can't give a specific reason why. My guess is that if you show the details of your work, it will become clear.

 

[dawg]

Please explain what you mean by this:

I have to say, though I haven't put any thought into the details, I would never expect your two questions to have related answers, though I can't give a specific reason why. My guess is that if you show the details of your work, it will become clear.

Dr Peterson, thanks for your response. That sentence is far from correct and now that I'm going over it so it my general description of the two problems. However, after correcting the calculations, my question regarding the intuition still stands. Attached is my solution for both problems, in the first, I calculate the expectation of the function of x: Y(X) = X/(1-X). In the second, I calculate the expectation of X and then evaluate the expression E[X]/((1-E[X]).

I understand that there is no formal reason for them to be the same (Y is not a linear function of X) and obviously the formal calculation gives the correct answers. However, if this experiment was conducted an infinite number of times, the average value of should converge to expectation, right ? If this is the case, and X converges to E[X], then the second value would give the average ratio. However, this would contradict the fact that the average value of Y over an infinite number of attempts would converge to the first value in the image below. I'm trying to gain some intuition as to why the results are different when there is a deterministic relationship between X and Y.

 

[Dr.Peterson]

However, after correcting the calculations, my question regarding the intuition still stands. Attached is my solution for both problems, in the first, I calculate the expectation of the function of x: Y(X) = X/(1-X). In the second, I calculate the expectation of X and then evaluate the expression E[X]/((1-E[X]).

I understand that there is no formal reason for them to be the same (Y is not a linear function of X) and obviously the formal calculation gives the correct answers. However, if this experiment was conducted an infinite number of times, the average value of should converge to expectation, right ? If this is the case, and X converges to E[X], then the second value would give the average ratio. However, this would contradict the fact that the average value of Y over an infinite number of attempts would converge to the first value in the image below. I'm trying to gain some intuition as to why the results are different when there is a deterministic relationship between X and Y.

In general, it is not true that the average of a function is the function of the average. You seem to be expecting that it should be for this function. Why? It seems to me that you just have a wrong intuition, and need to correct it.

The basic reason would be something you said yourself: integration over x is different from integration over y. In averaging the two ways, different weights are assigned to different cases (as in the classic problem of averaging speed when distances at each speed are the same vs. when times at each speed are the same).

 

[dawg]

In general, it is not true that the average of a function is the function of the average.

I understand, so although the average X would converge to E[X], the average of Y would not converge to E[X]/(1-E[X]), but to E[X/(1-X)], I guess I assumed the value of x would somehow be E[X] which would guarantee the value of y, but LLN only guarantees this about the average, my intuition was off as you say.
Thankyou