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I've got very much confused about distributions and am looking for quick help. Distributions are common in physics, so I humbly hope to receive an answer that will resolve my confusion.

Let's suppose we are conducting a physical experiment in which particles are born continuously at a constant rate. And let's suppose that once born, a particle lives a random time whose distribution is exponential with parameter $\lambda$. That is, the probability for a particle to die within an infinitesimal time interval $dt$ is $\lambda \, dt$ and doesn't depend on how much time has elapsed since the particle was born.

In this experiment, a stationary distribution of particles over their ages will be reached, and my understanding is that it will be an exponential distribution with the same parameter $\lambda$.

And since the distributions of ages and lifetimes are the same, the average age of living particles and the average lifetime must coincide (and be equal to $1/\lambda$).

But how is it possible if particles die after they live? How can the average age of living particles be equal to their average age at death?

My thoughts are running in circles in an attempt to resolve this, and I humbly hope that SE Physics users can shed light on what I'm missing.

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    $\begingroup$ Some live a short time, some live a long time. I'm not sure why you see a paradox here. $\endgroup$
    – Jon Custer
    Mar 16, 2023 at 13:35
  • $\begingroup$ "my understanding is...": are you claiming that this is true for any distribution? (It's not.) Or just for an exponential distribution? Can you provide a link? $\endgroup$
    – TonyK
    Mar 17, 2023 at 1:00
  • $\begingroup$ @TonyK My question is specifically about an exponential distribution, and my understanding is that for an exponential distribution the average age of living particles and the average lifetime at birth are the same. The question is how is that possible $\endgroup$
    – Mitsuko
    Mar 17, 2023 at 6:49

2 Answers 2

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The average lifetime of a particle "at birth" is $1/\lambda$. The average lifetime of the collection of particles that are alive at a given time is longer, because that is a biased sample. Indeed, since the average age of a living particle is $1/\lambda$ and the average remaining life expectancy of a living particle is also $1/\lambda$, the average lifetime of a living particle is $2/\lambda$.

This is like how for humans, the adult life expectancy is longer than the life expectancy at birth.

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  • $\begingroup$ Well now it is a paradox. A 70 y/o human is not the same as a 5 y/o age wise, but a 4B y/o U-238 atom dug up in a mine is the exact same as one that was made in a reactor today. $\endgroup$
    – JEB
    Mar 16, 2023 at 13:50
  • $\begingroup$ @JEB Keep in mind the 70-year-old's life expectancy is longer than that of the child, because despite the (likely) sooner death of the former, the life expectancy figure includes all of the time already lived. For U-238, the sooner death does not apply, but the inclusion of time already lived still does. $\endgroup$
    – Sten
    Mar 16, 2023 at 13:55
  • $\begingroup$ I don't care which one dies first, I'm just pointing out that a memoryless distribution is not like one that is not, and this site is full questions that are resolved when the OP understands that the age of an unstable particle/state has absolutely no affect on its future. $\endgroup$
    – JEB
    Mar 16, 2023 at 14:49
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For a particle with the given lifetime distribution, the stationary distribution of particles over their ages is not the same as the lifetime distribution.

Let's say the probability for a particle to be alive on time $t$ is $P(t)$. Then for a N particle system, the average number of particles of age $t$ is not $NP(t)$, instead, it is $NP(t)-NP(t+dt)$ for an infinitesimal time $dt$. $NP(t)$ actually includes all the particles which live longer than $t$.

As the stationary distribution of particles over their ages changes, the average age of particles at a stationary time will be different from the average lifespan of one particle.

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