Is average time between and after collision same for a gas?

[Linop]

I am stuck on this concept in my physics book where the author claims that in a low density ionic gas the average of the time between collision and average of the time taken from last collision in ions is same. He further states that the average time to the next collision is same as the average time from last collision. The velocity of each ion after collision is random.
Now if I apply definition of average of time intervals it follows from my calculations that average of time between collision is equal to the sum of average of time from last collision and to next collision. I am confused by this because the book specifically mentions that my result is incorrect. Please help me in this matter.
Ref: Electricity and Magnetism first edition by Edward M Purcell page# 121.
It is my first post in any physics forum so please forgive me for any mistakes.

 

[jbriggs444]

This is not a question about physics. It is a question about statistics and probability. The underlying problem is one of sampling bias.

If you pick a random time at which to examine the state of an ion then you will find that the "time to the previous collision" and the "time to the next collision" are both random variables with mean equal to the mean time between collisions. You are correct that this means that the "time between collisions" for this particular sample is the sum and is, on average, equal to twice the mean time between collisions.

This seems to be a conundrum. The duration of a randomly selected inter-collision interval is, on average, equal to twice the mean time between collisions?! How can this be so?

The problem is that this sampling process is biased. If we pick a random time to sample then we will tend to select long intervals more often than short. If we had, instead, picked a particular inter-collision free path to examine (say the one million and first), we would find that its duration is, on average, equal to the mean.

The latter idealized sampling procedure is how "mean time between collisions" is defined.

A bit of Googling yields... https://arxiv.org/pdf/1308.2729.pdf

Edit: Oh, and welcome to the forums. That was a good first post.

 

[Linop]

This is not a question about physics. It is a question about statistics and probability. The underlying problem is one of sampling bias.

If you pick a random time at which to examine the state of an ion then you will find that the "time to the previous collision" and the "time to the next collision" are both random variables with mean equal to the mean time between collisions. You are correct that this means that the "time between collisions" for this particular sample is the sum and is, on average, equal to twice the mean time between collisions.

This seems to be a conundrum. The duration of a randomly selected inter-collision interval is, on average, equal to twice the mean time between collisions?! How can this be so?

The problem is that this sampling process is biased. If we pick a random time to sample then we will tend to select long intervals more often than short. If we had, instead, picked a particular inter-collision free path to examine (say the one million and first), we would find that its duration is, on average, equal to the mean.

The latter idealized sampling procedure is how "mean time between collisions" is defined.

A bit of Googling yields... https://arxiv.org/pdf/1308.2729.pdf

Edit: Oh, and welcome to the forums. That was a good first post.

Thank you for your quick answer. I will be sure to study the pdf in the link. That problem was driving me crazy.