0
votes
1answer
55 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
0
votes
0answers
30 views

Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$ I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x} $$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
0
votes
0answers
10 views

Quadrivariate normal distribution?

Does anyone know how to write the probability density function of the Quadrivariate normal distribution? I was able to find the Bivariate here ...
1
vote
1answer
60 views

Intuition and the math behind normalization

What exactly is the purpose of normalization. From what I read, it is to adjust two different sets of values so you can compare them, but I don't understand why, nor the math behind it. Could anyone ...
0
votes
0answers
24 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
1
vote
1answer
76 views

How is the entropy of the multivariate normal distribution with mean 0 calculated?

Here is what I have so far: $$\begin{align} h(x) &= - \int \frac{1}{(2\pi)^{\frac{D}{2}}\det\Sigma^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T\Sigma^{-1}x) \ln ...
2
votes
0answers
91 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
2
votes
1answer
42 views

multi-variate normal distribution distance from vector sub-space

let $X\sim {\cal N}(\mu,C)$ be a random variable obeying multi-variate normal distribution in $\mathbb{R}^n$ and $U \subset \mathbb{R}^n$ be a vector space with $\dim(U)=n-1$. What is the probability ...
0
votes
1answer
73 views

Integral of Gaussian ring/shell

I would like to know the integral of the function $$f(\mathbf{x}) = {1 \over \sqrt{2\pi \sigma^2}} \exp\left\{- {(|\mathbf{x}| - \mu)^2 \over 2\cdot \sigma^2}\right\} $$ over an $n$-dimensional ...
1
vote
1answer
100 views

Sum of Gaussian Variables may not Gaussian

I am currently trying to understand the following three points which we discussed in lectures recently: We say that $X=(X_1,\ldots,X_d)$ is $d$-dimensional multivariate Gaussian distributed if ...
2
votes
2answers
93 views
0
votes
0answers
53 views

Transformation of multivariate normal distribution goes wrong

I have this distribution: $$\frac{e^{-\frac{\left(x-\mu _a\right){}^2}{2 \sigma _a^2}-\frac{\left(y-\mu _b\right){}^2}{2 \sigma _b^2}}}{2 \pi \sigma _a \sigma _b}$$ And integrating all over ...
1
vote
1answer
43 views

Proving MLE for normal distribution

I need to prove that using maximum likelihood estimation on both parameters of normal distribution indeed maximises likelihood function. So, the log-likelihood function for parameters $\sigma$ and ...
0
votes
0answers
93 views

Multivariate Normal Product Distribution

I am looking for multivariate case of a distribution of a product of two normally distributed variables X and Y. The variables are independent. Something similar to this: ...
2
votes
1answer
400 views

What is the analytic expression for PDF of joint distribution of two Gaussian random vectors?

I know that if $X$ and $Y$ are random variables with respective PDFs, $$ f_X(x) = \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left\{-\frac{\left(x-\mu_x\right)^2}{2\sigma_x^2}\right\} \\ f_Y(y) = ...
1
vote
1answer
816 views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
0
votes
1answer
29 views

How to generate this vector?

I have two vectors $A$, $B$ each containing 10 random numbers from standard normal distributions. I want to generate another vector $C$ of $10$ numbers from standard distribution where ...
4
votes
1answer
2k views

Multivariate Normal Difference Distribution

Since the distribution of a difference of two normally distributed variates X and Y with means and variances $(\mu_x,\sigma_x^2)$ and $(\mu_y,\sigma_y^2)$ respectively is given by another normal ...
2
votes
3answers
242 views

finding unknown variable in Gaussian Integral

Given values of d, p and $\sigma$, is it possible to calculate the value of $\mu$? $$1-\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{y-d}^{y+d}\exp\big(-{x^2}/{2\sigma^2}\big) ...
3
votes
2answers
381 views

Nested normal-distribution integral

Is there an analytical or approximate solution of the following integral? $$ \int_{-\infty}^{\infty}\int_{y-d}^{y+d}\exp\big(-{(x-\mu_1)^2}/{2\sigma^2}\big) \exp\big(-{(y-\mu_2)^2}/{2\sigma^2}\big) ...