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In my class we have the following equation.

$\displaystyle Pr[N(t + ∆t) − N(t) = k] = \frac{e^{\lambda\Delta t} }{ k!}$, where $k = 0, 1, 2, \ldots \quad$ $(1)$

We are talking about the poisson process. I understand what the LHS means of $(1)$ and also what the RHS means.

What I don't understand is the following approximation that is used to prove further results:

$e^{\lambda\Delta t} \approx 1 + \lambda\Delta t + o(\Delta t)$ $\quad (2)$

Could someone explain why this is true?

Thank you for your help!

jdods
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Joe
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1 Answers1

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we have

$$exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$

when $x$ is small, the higher order terms vanishes first.

Side remark: is there a typo in the question, should the sign before $\lambda \Delta t$ be positive on the RHS? or should there be a negative in the exponential on the LHS?

Siong Thye Goh
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  • The way I had it before is how I have it in my lecture notes...although they are known to be peppered with errors throughout (the lecture notes are distributed by my prof at the beginning of the semester). So I guess the above correction is right? – Joe Apr 28 '16 at 03:57
  • seems to make sense to me. =) – Siong Thye Goh Apr 28 '16 at 03:58
  • Ok, so I just did out the summation and I see what you mean that the higher order terms vanish first. So are all values where n>2 captured in the function o($\Delta t$)? – Joe Apr 28 '16 at 04:11
  • yup! that is right. – Siong Thye Goh Apr 28 '16 at 04:16
  • hey, I look at the question again, and I think equation 1 might not be accurate. within a time interval of $\Delta t$, it should follows a Poisson distribution. $\frac{(\lambda \Delta t)^k}{k!}\exp(-\lambda \Delta t)$. – Siong Thye Goh Apr 28 '16 at 08:00
  • Hey I think you are right. Given that you are correct would e^(-lambdadelta[t]) ~= 1 - (lambda)(delta)[t] + o(delta[t])?

    Sorry for formatting again, finals in a few days and can't spend time learning how to latex right now! But soon I will.

     – Joe Apr 30 '16 at 18:12
  • yup, that is true. – Siong Thye Goh Apr 30 '16 at 18:30
  • Thank you so much for clearing that up. – Joe Apr 30 '16 at 18:34