Derivation of the exponential distribution - that infinitesimal

Click For Summary
SUMMARY

The discussion centers on the derivation of the exponential distribution, specifically focusing on the role of the little-o notation, denoted as o(Δt). This notation encapsulates all higher-order infinitesimal terms, such as (Δt)² and (Δt)³, which are not significant in the first-order Taylor approximation. Participants confirm that this approach simplifies the derivation by neglecting terms that do not contribute to the linear approximation of the exponential function, e^x, when considering small changes in Δt.

PREREQUISITES
  • Understanding of little-o notation in calculus
  • Familiarity with Taylor series and Taylor expansions
  • Basic knowledge of exponential functions and their properties
  • Concept of infinitesimals in mathematical analysis
NEXT STEPS
  • Study the application of Taylor series in approximating functions
  • Explore the implications of little-o and big-O notation in mathematical analysis
  • Learn about the properties and applications of the exponential function, e^x
  • Investigate higher-order approximations and their relevance in calculus
USEFUL FOR

Mathematicians, statisticians, and students studying calculus or mathematical analysis, particularly those interested in the properties and derivations of exponential functions.

thomas49th
Messages
645
Reaction score
0
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
 
Physics news on Phys.org
##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
 
why do those terms exist? was there some kind of taylor expansion?
 
Exactly. It is a taylor approximation to first order (linear in Δt) and higher orders are neglected.
 
what the taylor expansion of? e^x ?
 
Should be ##e^x - e^{x+Δt}## with some prefactors I did not check.
 

Similar threads

Undergrad Beta Distributed Random Variates
  • · Replies 4 ·
Replies
4
Views
2K
Graduate How Is the CDF Expressed for a Uniform Distribution on a Circle?
  • · Replies 8 ·
Replies
8
Views
2K
Graduate What exactly is a "rare event"? (Poisson point process)
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Variable transformation for a multivariate normal distribution
  • · Replies 1 ·
Replies
1
Views
2K
MHB Normal and exponential distribution
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Elementwise Derivative of a Matrix Exponential
  • · Replies 2 ·
Replies
2
Views
3K
High School How do I invert this exponential function?
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Why is the derivative defined as the limit of the quotient?
  • · Replies 13 ·
Replies
13
Views
2K
How Do You Find the Density Function of a Sum of Exponential Random Variables?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Finding the derivative of a characteristic function
  • · Replies 5 ·
Replies
5
Views
3K