What is the Origin of the e-Term in the Characteristic Function?

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Homework Help Overview

The discussion revolves around the origin of the e-term in the characteristic function, specifically in the context of probability and random processes. Participants express confusion about the transition from discussing distributions and expected values to the introduction of the characteristic function.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants question the source of the e-term in the characteristic function and seek to understand its relevance. There is a discussion about the notation differences between engineering and mathematics, particularly the use of "j" versus "i". Some participants express a desire for a broader understanding of the connections between concepts.

Discussion Status

The discussion is ongoing, with participants sharing thoughts and questioning assumptions. Some guidance is offered regarding the notation used in different fields, but there is no clear consensus on the origin of the e-term or its implications for understanding the characteristic function.

Contextual Notes

Participants note that the textbook does not provide a clear explanation for the e-term, leading to confusion. There is an acknowledgment of the differences in terminology and notation between engineering and mathematics, which may contribute to misunderstandings.

EvLer
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This kind of bothers me:
our textbook does not explain (and the professor either) where characteristic function comes from, all it says is what it defined as, which is E[ejwX], where E is expectation of random variable X. But where is this e-term coming from?

Thanks in advance.
 
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e is, well, e, you know, euler and all that? Logs base e...
 
That's clearly an engineering text since they are using "j" where any normal human being would use "i"!

Although you don't say it, I suspect you are talking about the average value of waves with ejwX= eiwX= cos(wX)+ i sin(wX).
 
HallsofIvy said:
That's clearly an engineering text since they are using "j" where any normal human being would use "i"!
yeah... i heard that argument before: math people are i-people, engineers are j-people, forgot what the reasoning is...i think engineers underestimate "i" in some way...:rolleyes:
Although you don't say it, I suspect you are talking about the average value of waves with ejwX= eiwX= cos(wX)+ i sin(wX).
that's the thing! i know the Euler's identity and all that, but the book is "Probability and random processes for electric engineers", and all of a sudden after discussing various distributions, pdf-s, cdf-s we have expected value of a random variable... so far so good, THEN... we get into n-th moment and so here we go... characterstic function and n-th moment theorem. From the book, we can find n-th moment either by integral or by differentiating this characteristic function (couple other things to that), so my question is: how the heck did they come up with that! I'm just trying to make connections and see some continuity to all this... (signals are not in the picture, at least yet...) math is not the issue, i can deal with that...the idea "why and how" is the issue.
So... if someone could give me the "big-picture" stuff...:rolleyes:
 
Last edited:
EvLer said:
yeah... i heard that argument before: math people are i-people, engineers are j-people, forgot what the reasoning is...i think engineers underestimate "i" in some way...:rolleyes:

Engineers use i to stand for electrical current.
 
Corneo said:
Engineers use i to stand for electrical current.
Which causes enormous troubles when working with jmaginary numbers!
 
Corneo said:
Engineers use i to stand for electrical current.
hm... really? :)
i was getting at something else actually, it has something to do with vector properties of i vs j, as a vagually remember...
 

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