Is ln(E(x^n)) equal to E(ln(x^n))

  • Thread starter Thread starter xuej1112
  • Start date Start date
Click For Summary

Homework Help Overview

The discussion revolves around the relationship between the natural logarithm of the expectation of a function and the expectation of the natural logarithm of that function, specifically in the context of continuous variables and the properties of expectation.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of expectation and logarithmic functions, questioning the equality of ln(E(x^n)) and E(ln(x^n)). There is also a discussion about the implications of Jensen's inequality in relation to concave and convex functions.

Discussion Status

Some participants have provided clarifications regarding the definitions involved, while others have pointed out the general inequality properties of expectations. The conversation is exploring different interpretations of the mathematical properties without reaching a consensus.

Contextual Notes

There is an ongoing discussion about the nature of the functions involved and the assumptions regarding the continuity and non-negativity of the variable x. The distinction between concave and convex functions is also being examined in the context of Jensen's inequality.

xuej1112
Messages
17
Reaction score
0
is ln(E(x^n)) equal to E(ln(x^n)), if x is a continuous variable and x>=0?

Help me! Thanks!
 
Physics news on Phys.org


This would more appropriately have been placed in the Calculus and Beyond homework help section, even if it isn't homework it is of that nature and level. General Math is more suited for general discussion of mathematical ideas.

Anyway, what is E? Is it the exponential function? How are the exponential function and natural logarithm related.
 


Sorry. E is the expectation
 


Oh ok. In that case, no it's not true. In fact, in general E(g(x)) is not equal to g(E(x)) which is due to the definition of E(x) as an integral with x as the integrand. This makes it inherently difficult for these nice properties to occur, as integrals don't have "nice" properties past linearity. So while E(aX+bY) = aE(X) + bE(Y), E(X^n) is not anything nice in terms of E(x).
 


thanks. if g is a concave function, can i say g(E(x))>=E(g(x)) ? (jensen's inequality)
 


It's the other way around, g(E(X)) is LESS or equal to E(g(X)), if g is convex (not concave).
 

Similar threads

##\lim_{n\to\infty}\left(n\left(1+\frac1n\right)^n-ne\right)=?##
  • · Replies 5 ·
Replies
5
Views
2K
Integration problem using inscribed rectangles
  • · Replies 14 ·
Replies
14
Views
2K
Proving limits for roots and exponents
  • · Replies 13 ·
Replies
13
Views
2K
Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x
  • · Replies 34 ·
2
Replies
34
Views
4K
Please check my calculations with these complex numbers
  • · Replies 11 ·
Replies
11
Views
1K
Find the equation of given curve in the form ##e^{3y}=f(x)##
  • · Replies 4 ·
Replies
4
Views
2K
Integrating Logarithmic Functions with Binomial Terms
  • · Replies 14 ·
Replies
14
Views
3K
Please verify my answer to this question about the domain
  • · Replies 2 ·
Replies
2
Views
1K
Proving convergence and divergence of series
  • · Replies 3 ·
Replies
3
Views
2K
Solve the given equation using the Lambert W function
  • · Replies 2 ·
Replies
2
Views
2K