# Tagged Questions

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### Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
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### the characteristic function of this distribution is equal to 0 everywhere except at the origin, mistake?

I wanted to compute the characteristic function of the distribution in question here: How to multiply a standard normal RV times a uniform{-1.1} RV? Let $X$ be standard $N(0,1)$, $Y$ be Uniform ...
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### Characteristic function of an integer-valued distribution, inversion formula

I am working on the following: Show that if $\varphi$ the characteristic function of an integer-valued distribution then \begin{align*} \mathbb P(X = k) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-itk} ...
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### Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
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### Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
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### How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
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### Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
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### Conditional Characteristic Function

Given I know the joint characteristic function of the random variables $X,Y$ and the characteristic function of $Y$, is there a way to recover the characteristic function of $X|Y$ without inverting ...
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### Does “independence” of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how ...
I need to find the probability density of some distribution with characteristic function given by: $$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$ I know the formula for inverting a ...