1

We know that Eulers number $e$ can be expanded as $$e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} ...$$

An extended definition for the powers of $e$ are: Equation 2

$$e^\lambda = \sum_{k=0}^{\infty}{\frac{\lambda^k}{k!}} $$

Which is something I learned recently in statistics, as it is used in the Poisson pmf.

I was hoping to find a nice story for how this ties into my understanding of Poisson distribution or Euler's number or what trickery was used to prove this infinite series as the common method for calculating the value of an infinite series by substitution fails here. Does anyone have a reference/proof of this infinite series expansion of the powers of $e$?

Edit: as suggested let me add context to narrow down the question and move it more towards stats than math oriented

The pmf of a distribution must sum to 1. The Poisson pmf is defined as the RHS of equation 2 above. To normalise it, it must be divided by a normalisation factor, which happens to be $\frac{1}{e^\lambda}$ I am trying to understand how the RHS which is well understood as the Poisson events, happened to sum to the LHS, and if there is causation there or just coincidence. Because if it is coincidence and we normalise every PMF value by a constant that is slightly off, it would screw up the entire pmf. And Poisson is used to work with small probability values and computationally the probability checks out, but there is nothing better than proof by numerical computation to some decimals?

devssh
  • 111
  • 4
  • There's no trickery: many math texts, from the most elementary to the most advanced, *define* $e^\lambda$ in terms of this series. – whuber May 13 '20 at 12:29
  • Yes, but if it is so common there must be a proof somewhere/anywhere in the world? is this an axiom... because everything here is real and discrete. no imaginary or complex numbers, simple numerical additions. And convergence to a simple non-recurring decimal – devssh May 13 '20 at 12:31
  • So that makes for easy explanations and practical applications in the real world but I couldn't find proof. – devssh May 13 '20 at 12:37
  • 2
    What are you asking about? The series expression for $e^\lambda$ is treated in any introductory calculus textbook, and it's off topic here (you could try Math.SE). Or are you wondering about any relationship to the Poisson distribution? Please clarify. – Stephan Kolassa May 13 '20 at 12:50
  • 3
    If you want a proof from first principles that $\exp(\lambda)=e^\lambda$ then you will need (a) a definition of $\exp(\lambda)$, (b) a definition of $e$, and (c) a definition of $a^b$ where $a$ is a positive real and $b$ is any real. Can you provide these? – Henry May 13 '20 at 13:34
  • Suggestions will be subjective. I personally fell 3blue1brown's videos are very helpful [Euler's formula with introductory group theory](https://www.youtube.com/watch?v=mvmuCPvRoWQ) [What's so special about Euler's number e](https://www.youtube.com/watch?v=m2MIpDrF7Es) [Understanding e to the i pi in 3.14 minutes](https://www.youtube.com/watch?v=v0YEaeIClKY) – Haitao Du May 13 '20 at 12:18
  • But there must be a reference/proof or original paper of someone who proved this? Something less broad and more specific to the equation I needed proof for? – devssh May 13 '20 at 12:21
  • since I'm more interested in $e^k$ where k is real, not imaginary like an amalgamation of $pi$ and complex $i$ – devssh May 13 '20 at 12:29
  • I see, $a^b$ is a+a...(b times). The definition of $e$ is provided in the question. As for exp($\lambda$) I think that's not what we are after. I'll edit the question and add an example. – devssh May 14 '20 at 18:50
  • Ah I see what you mean @Henry , just the thought of$e^2$ is hard to define in addition because it's 2.71*2.71 ... @ StephanKolassa you might be right, this needs to be asked on a math forum if it's about the proof of something that is fundamentally treated as axiom in stats textbooks – devssh May 14 '20 at 19:21
  • A *statistical* development of the ideas in your question can be found at https://stats.stackexchange.com/questions/214421. It shows the connection between Poisson probabilities and exponentials is not accidental. – whuber May 14 '20 at 19:24

0 Answers0