0
votes
0answers
32 views

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}] = ...
0
votes
1answer
47 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
2
votes
2answers
53 views

Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$

I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$ where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
2
votes
0answers
53 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
0
votes
0answers
56 views

Fourier transform of given characteristic function

I have a function $$g(l) = E [ e^{iuX}|X>l ] - Prob (X>l) $$ and i need to derive how its Fourier transform is: $$F_{l,v}(g(l)) = \frac{\phi_X(u+v)-\phi_X(v)}{iv}$$. This gets down to ...
2
votes
1answer
96 views

Fourier transform of characteristic function in a sphere

A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ ...