Derivation of the exponential distribution - that infinitesimal

[thomas49th]

Hello,

I've been looking at the derivation of the exponential function, here
http://www.statlect.com/ucdexp1.htm
amongst other places, but I don't get how, why or what the o(delta t) really does. How does it help?

It's really confusing me, and all the literature I've looked at just seems to quickly dance over it

Thanks

 

[mfb]

$o(\Delta t)$ contains all small terms the derivation does not want to (and does not have to) care about. Something like $(\Delta t)^2$, $(\Delta t)^3$ and so on.

Little-o notation

 

[thomas49th]

why do those terms exist? was there some kind of taylor expansion?

 

[mfb]

Exactly. It is a taylor approximation to first order (linear in Δt) and higher orders are neglected.

 

[thomas49th]

what the taylor expansion of? e^x ?

 

[mfb]

Should be $e^x - e^{x+Δt}$ with some prefactors I did not check.