Need to prove that the following integral is positive

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Discussion Overview

The discussion revolves around proving that a specific double integral is positive. The integral involves a normal distribution function and an exponential decay term, with participants exploring the implications of the function's properties and the nature of the integral.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant requests assistance in proving the positivity of the integral without providing details about the function f.
  • Another participant notes that the positivity of the integral depends on the nature of f, which has not been specified.
  • A participant mentions that f is normally distributed and has calculated the inner integral in terms of the error function, expressing hope for a closed form solution.
  • One participant asserts that since all components of the integral are positive functions, the integral should be positive if it exists, and suggests that it can be shown to exist through basic approximations.

Areas of Agreement / Disagreement

Participants express differing views on the integral's positivity, with some asserting it is positive based on the properties of the functions involved, while others emphasize the need for more information about f to draw any conclusions.

Contextual Notes

The discussion highlights the dependence on the specific form of the function f and the assumptions regarding the integral's convergence and existence.

ardhu
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I need help proving that the following (double) integral is positive:

int(limits from -infinity to +infinity) f(b) int(limits from b to +infinity) [e^{-(x-mean)/(2 *variance)}]/sq root(2 *pi *variance) dx db

thanks
 
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This would be more appropriate elsewhere (calculus, say), but since exp is continuous and always strictly positive it would entirely depend on what f is, and you've not stated that.
 
I moved this thread to the Calculus homework forum, and deleted ardhu's duplicate thread in another forum. ardhu -- do not double-post, and please post homework questions in the appropriate homework forum. You also need to show your work so far in order for us to help you (Physics Forum rules).
 
f is distributed normal. a quick update- i did calculate the inner integral in terms of the error function. and as the f is gaussian, I'm hoping that a closed form solution will emerge.i just need a clever way to prove that the net result is positive.

also this is not a homework problem.
 
ardhu said:
also this is not a homework problem.

Fair enough, I'll take you at your word on that. I've moved the thread to the forum that matt suggested. Welcome to the PF, BTW.
 
Everything in sight is a positive function, so the integral, if it exists, is positive. And the integral is easily shown to exist by some naive approximation (pretty much anything - it decays exponentially).
 

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