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Consider a birth process, where at each unit time step $t$, on the average, there is a fraction of $0.012$ among all the populations who will give a birth. Suppose the doubling time is $m$, then, from $1.012^m =2$ we get that $m$ is approximately $58.108$, i.e., the whole population will double after $58.108$ time steps. Then, the frequency, i.e., the average rate of events per time step is $1/58.108$. We can thus model it using a possion process with parameter $1/58.108$. Then, the probability for a birth-event within one time step is $1-e^{-1/58.108}$, which is approximately $0.0171$. This reasoning should be right, right?

Now, probability is a proportion in some sense. Note the fact that a proportion of $0.012$ would give a birth during each time step. So, is it safe to say that the probability of giving a birth during each time step is $0.012$ then? But that is not the same as $0.0171$, not even close. Can anyone tell me why? Any comments are greatly appreciated.

Stanley
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nstrong
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1 Answers1

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You need to be careful with your definition, because you've defined it as a discrete time process where individuals have either $1$ or $0$ births in each interval. But you're trying to model it as a continuous time process. Which is it?

Even if it is a continuous time process, it still doesn't make sense to use a time-homogeneous Poisson process (as you did, giving it a constant rate $1/58.108$). You will need to convince yourself that if individuals are each giving birth at a rate of $0.012$, then a larger population will have a larger population birth rate.

The thing which does this is called a pure birth process, or Yule process. This is different from a time-homogeneous Poisson process (constant rate $\lambda$) and different also from a time-inhomogeneous Poisson process (rate $\lambda(t)$ depends on time). In the pure birth process, the rate $\lambda_n$ depends on the number of individuals in the population. If you let the rate be equal to $\lambda \cdot n$ (i.e. a constant multiple of the population size) then the result is that the population size $N(t)$ is, unfortunately, not even a Poisson random variable. It turns out that $Pr[N(t) = n | N(0) = k]$ has a negative binomial distribution with parameters $k$ and $e^{\lambda t}$. Hope this helps!

Stanley
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  • Just to make sure, the poisson process is continuous in nature, and its discrete-counterpart is the same as the sum of bernoulli process, is it right? – nstrong May 21 '15 at 16:15
  • What is the difference between discrete and continuous poisson process? Even it is contious, I can still model it step by step, right? – nstrong May 21 '15 at 16:52
  • A Poisson process is continuous by definition and its discrete analog is the Bernoulli trials process. For more discussion about this look here.

    You can model something that is continuous using a discrete 'step by step' process, but you'd have to recognise this as a simplifying assumption.

     – Stanley May 21 '15 at 20:36
  • One question, when you said " if it is continuous time process, it does not make sense to use a time-homogeneous poisson process.", you mean " if it is NOT continuous time process, ...", right? – nstrong May 21 '15 at 20:39
  • Sorry, that was ambiguous. If it is a discrete time process then it's probably inappropriate to use a time-homogeneous Poisson process. If you decide it is a continuous time process, you would still be mistaken to use a time-homogeneous Poisson process. – Stanley May 21 '15 at 21:07
  • I am confused then. So, what is the reason for not using time-homogeneous poisson process if it is continuous? – nstrong May 21 '15 at 21:13
  • The reason is that over time the population becomes larger and the rate of birth (in the whole population) increases. – Stanley May 21 '15 at 21:57
  • I thought the rate of birth (the parameter \lambda) is constant through the whole process. So if each time the number of new born people is increasing, it would somehow violate the definition of time-homogeneous poison process? – nstrong May 22 '15 at 01:28
  • Yes. Exactly. Do you understand why the birth rate is increasing? – Stanley May 22 '15 at 06:48
  • No. Why is the birth rate increasing? I understand each time the number of new born is increasing, but the birth rate is the constant value \lambda, right? – nstrong May 22 '15 at 10:52
  • Let us continue this discussion in chat. – Stanley May 22 '15 at 11:10
  • I cannot log in to chat, because I do not have 20 reputations right now. My question is, when you talk about the rate, you actually mean the number of new born people each time, right? – nstrong May 23 '15 at 11:40
  • The formal definition is that the birth rate $\lambda(t)$ is such that the probability of a new birth in a small interval $(t, t+h)$ is $\lambda(t) \cdot h + o(h)$ where $\lim_{h \rightarrow 0} \frac{o(h)}{h} = 0$. Informally, it is the expected number of new births per time unit at time $t$. So I'm claiming that this $\lambda(t)$ depends only on the number of individuals in the population at time $t$, which is of course random. – Stanley May 23 '15 at 12:39
  • One quick question, if we have one single population in the beginning, then then number of populations at time t would be geometric, right? – nstrong May 28 '15 at 14:02
  • So, yule process is discrete or continuous? From the definition of birth rate you gave, it should be continuous, while the geometric distribution is discrete. – nstrong May 28 '15 at 17:35
  • The population size is discrete isn't it? This is analogous to the Poisson process, where waiting times are exponential (continuous) but the event count is Poisson (discrete) – Stanley May 28 '15 at 20:01