Articles of gaussian integral

Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$

For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal distribution $x \sim N(0,\Sigma)$? I have difficulty to carry out the integral, but would guess the result is related to the norm of $\Sigma$.

How to integrate $\int_0^{\infty}\frac{e^{-(t+\frac1t)}}{\sqrt t} dt$?

This is a problem given in my homework . I have to find the integral$$\int \limits_{0}^{\infty} \frac{e^{-(t+\frac{1}{t})}}{\sqrt t}dt$$ I am trying to use integral representation of the gamma function but I was not able to get it in the region of convergence i.e. $\int \limits_{0}^{\infty} \frac{e^{-t}}{\sqrt t}$ is clearly $\Gamma (\frac{1}{2})$ but the second factor […]

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)

We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$ We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann integral not Lebesgue integral. Here $D_{r}$ is the disc of radius $r$ with respect to the Euclidean norm of $\mathbb{R}^{n^{2}}$. Is the above integral a convergent improper integral? What about if we consider $D_{r}$ with […]

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?

Is the following integral a convergent integral? Can we compute it, precisely? $$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$ Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq \mathbb{R}^{n^{2}}$? So $\mu$ can be counted as $\mu=\prod_{i,j} da_{ij}$ Note: If this integral would be convergent , either in Lebesgue or in Riemann sense, then it would be equal to a scalar […]

Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$

I have to find $$I=\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx $$ I think we could use $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2} $$ But I don’t know how. Thanks.

Evaluating $\int_0^\infty x^2e^{-\alpha x^2}dx$ and $\int_0^\infty xe^{-\alpha x^2} dx$ knowing $\int_0^\infty e^{-\alpha x^2}dx$

As the title, question 5 in this picture thanks

Proof of a Gaussian Integral property

I’m working through some old integrals and I found one that’s interesting. I can’t quite remember how it’s proved, so if someone could set me off in the right direction, it would be really helpful. Thanks! $\int_{\mathbb{R}} e^{-ax^2 + bx} dx = \sqrt{ \dfrac{\pi}{a}} e^{\frac{b^2}{4a}}$ I’ve tried change of variable, but I’m not sure this […]

How to compute the integral $\int_{-\infty}^\infty e^{-x^2/2}\,dx$?

Yes, I know that this is very similar to $\int_{-\infty}^\infty e^{-x^2}\,dx$, which has been answered a million times, but I still don’t know how to apply the technique from that integration to mine. I don’t want to do this using polar coordinates, or “erf”. I’d like to use the Gamma function (which I assume is […]

Convolution with Gaussian, without dstributioni theory, part 3

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ u(t,x)=\int_{\mathbb{R}^n}\Gamma(t,x-y)f(y)dy. $$ If $p=\infty$, then $u(t,\cdot)\to f$ a.e. as $t\to 0$. I’ve shown that $||u(t,\cdot)||_p\leq ||f||_p$ for $1\leq p\le\infty$ but haven’t gotten anywhere from here. Could someone please give some hints […]

Compute complex Gaussian integral

I don’t know how to work out the homework of Leib&Loss P121, Ex4(b), in which we need to compute the following $$ \int_{R^n}\exp(-x^tAx)dx=\pi^{n/2}/\sqrt{\det A} $$ where $A$ is a symmetric (thank Paul, see the comments) complex matrix with positive definite real part. It hint to use something like continuous extension, but I don’t know how […]