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

[EvLer]

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[e^jwX], where E is expectation of random variable X. But where is this e-term coming from?

Thanks in advance.

 

[matt grime]

e is, well, e, you know, euler and all that? Logs base e...

 

[HallsofIvy]

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 e^jwX= e^iwX= cos(wX)+ i sin(wX).

 

[EvLer]

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...

Although you don't say it, I suspect you are talking about the average value of waves with e^jwX= e^iwX= 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...

 

[Corneo]

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...

Engineers use i to stand for electrical current.

 

[HallsofIvy]

Engineers use i to stand for electrical current.

Which causes enormous troubles when working with jmaginary numbers!

 

[EvLer]

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...