Individuality of Characterstic Functions in Probability

According to Wikipedia, a particular function entirely establishes the buildings of a probability circulation. This suggests it has to be one-of-a-kind. Nonetheless, the definition offered is:

$$ \text{Char of }X (t)=E[e^itX]$$

Currently $e^{iz}$ repeats for every single $2 \pi$ increase in $z$. So just how can it be one-of-a-kind?

2019-05-06 01:52:56
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Answers: 2

It's simply a Fourier change. E (x) is an indispensable over the probability circulation. The function is one-of-a-kind ; if you concentrate on the value inside the assumption indispensable, that's not, yet so what?

2019-05-08 13:36:04

I assumed in the beginning you were asking just how the particular function can be one-of-a-kind. There's no concern below, due to the fact that e ^ ix is a well - specified function : for an offered value of x, there is an one-of-a-kind value of e ^ ix. Therefore the assumption (an indispensable) has an one-of-a-kind value too.

On 2nd idea, it appears you're asking just how it's feasible, taking into consideration that e ^ ix is not injective (i.e., numerous x can have the very same value of e ^ ix), that the initial circulation can be entirely recouped from the particular function. The response to that is that you're possibly missing out on the "t" in the expression : the particular function is a function of t, and also although for a offered value of t (as an example t = 1), the arbitrary variables $X$ and also $X+2k\pi/t$ would certainly have the very same value of the particular function then, they would certainly have various values at various other t. So the particular features are various, and also of course, the circulation can be entirely recouped from the particular function.

2019-05-08 13:21:43