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probability theory - Characteristic function and moments
2017年10月9日 · It is known that the characteristic function of a random variable can be used to find the moments. We have that $\frac{d\phi^{(k)}(t)}{dt}|_{t=0}=i^kE(x^k)$, so given a sequence of moments, how can that information be used to recover the characteristic function?
Characteristic function of a standard normal random variable
2011年11月27日 · I will give two answers: Do it without complex numbers, notice that $$ \begin{eqnarray} \mathcal{F}(\omega) = \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} \mathrm{e ...
measure theory - Why the characteristic function is measurable ...
$\begingroup$ @User0.618 Here $\{0,1\}$ is the range in its definition, because the codomain may differ in the context, for example one could also look at the characteristic function in $\mathbb{C}$. $\endgroup$ –
Proving that the characteristic function is uniformly continous.
I am trying to prove that the characteristic function is uniformly continuous. I understand how to get to this bound: And I would like to find the $\delta$ as a function of $\epsilon$ but I am used to having $|f(x) -f(c)|$ instead of $|f(x+h) - f(x)|$ when doing uniform continuous proofs.
Characteristic function of exponential and geometric distributions
I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you please guide me on getting them? Here is my solution so far: Characteristic function of exponential distribution:
statistics - Characteristic function of the normal distribution ...
Characteristic function of standard normal distribution using this method. 0. Trick to solve an improper ...
Why does the characteristic function always exist?
2015年3月20日 · I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a given probability distribution. Given the the moments don't necessarily exist, why is it that the characteristic function still exists?
How to get PDF from characteristic function
2013年9月16日 · $\begingroup$ Hey , I know comment is quite late , but it would be really helpful if you could tell $ f(x) $ of characteristic function = $( {1- |t|})^{-1}$ $\endgroup$ – simran Commented Jun 29, 2023 at 3:52
Real-valued characteristic function of $X \\sim U[-1, 1]$
In the Wikipedia entry for characteristic function there is this plot: with the caption: The characteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued.
real analysis - Derivatives of characteristic function - Mathematics ...
Characteristic function and moment generating function: differentiating under the integral. 2. When can ...