# 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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### Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
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### Inverting a difficult characteristic function.

Let $X$ be a random variable with characteristic function $$\phi_X(w)= \frac{1}{1-\frac{iw}{\lambda}e^{(\lambda-iw)\eta }}$$ where $\lambda$ and $\eta$ are constants. What is the pdf of $X$?
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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 Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
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### Show that $\frac{1}{n}\sum_{j=1}^{n}X_{j}$ is Cauchy distributed when the $X_{i}$ are all Cauchy

Let $X_{1}, \cdots, X_{n}$ be i.i.d. Cauchy random variables with parameters $\alpha=0$ and $\beta=1$. (That is, their density is $f(x)=\frac{1}{\pi\,(1+x^{2})}$, $-\infty < x < \infty$.) Show ...
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### Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...