# 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?

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.

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