Questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution.

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6
votes
0answers
233 views

What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
5
votes
0answers
76 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
4
votes
0answers
128 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
3
votes
0answers
34 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
3
votes
0answers
61 views

Independent normal distributions

I found two theorems with a similar content and want to find out which one is true: Let $X,Y$ be normally distributed random variables and $X+Y$ is also normally distributed or $ (X,Y)$ is ...
3
votes
0answers
27 views

Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt $$ Let $F_{Y_n}$ ...
3
votes
0answers
60 views

Length of Gaussian distributed variables

Suppose I have a set of random variables $x_1,...,x_n$ s.t. $x_i\sim N(\bar{x}_i,\sigma_i^2)$. And I define a new variable $x=\sqrt{x_1^2+...+x_n^2}$, then will $x$ also be normally distributed? And ...
3
votes
0answers
88 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
3
votes
0answers
76 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
3
votes
0answers
685 views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
3
votes
0answers
145 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
3
votes
0answers
66 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
0answers
84 views

Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want ...
2
votes
0answers
20 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
2
votes
0answers
51 views

On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$

Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
2
votes
0answers
29 views

How to test a hypothesis which compares set of pairs of statements?

I've conducted an experiment but I'm not sure how to proceed with statistical analysis of it. I have pairs of sentences created by two groups of people A and B, semantically the sentences in each ...
2
votes
0answers
25 views

Variance of a Population of Two Indpendent Random Variables

I have a question regarding a problem I'm looking at out of personal curiosity. Here is the basic setup of the problem: There is a population that contains half of type A, and half of type B. The ...
2
votes
0answers
48 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...
2
votes
0answers
75 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
2
votes
0answers
97 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
0answers
86 views

Maximum Posterior: $ p(\bf{w}\mid\bf{x},\bf{t},\alpha,\beta) \propto p(\bf{t}\mid\bf{x},\bf{w},\beta)p(\bf{w}\mid\alpha) $ for Gaussian Distribution

At the moment I take a look at the book Pattern Recognition and Machine Learning from Christopher Bishop and as I try to understand the basics of the probability theory I get stuck trying to ...
2
votes
0answers
38 views

Mixture of Gaussians — Distribution Weight

I've been having trouble understanding how to simplify (as well as understand) the equation for what I'm calling the "Distribution Weight" of a Conditional Mixture of Gaussians distribution. Namely, ...
2
votes
0answers
94 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
2
votes
0answers
32 views

Combining two circulating normal distributions

I am working in estimating the impact of location error on location based services. My question is listed below. If the error distribution of location estimation follows in general a normal ...
2
votes
0answers
114 views

Getting a Hermite polynomial expansion of Gaussian with given variance.

I am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate ...
2
votes
0answers
285 views

Distribution of the $l_2$-norm of gaussian vector

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution. What is the distribution in the general case? ...
2
votes
0answers
32 views

The distribution of the result of Monte-Carlo method

For example, if I want to determine the probability of getting tails when tossing a coin. By Monte-Carlo method, I toss the coin 1000 times and got 600 tails. As I know the distribution of the result ...
2
votes
0answers
27 views

Multivariate Distribution Question?

If $(X,Y)$ have the following joint distribution: $$f_{X,Y}(x,y) = \begin{cases} 2 f_X(x)f_Y(y) & \text{if }xy>0 \\[6pt] 0 & \text{otherwise} \end{cases} $$ where $f_X(·)$ and $f_Y(·)$ ...
2
votes
0answers
70 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{ ...
2
votes
0answers
826 views

Standardized Normal Distribution Problem

Mopeds (small motorcycles with an engine capacity below $50~cm^3$) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the ...
2
votes
0answers
98 views

Can any one help me normalize this equation? (Modified 3D Gaussian)

$$\exp\left( - e^{d-sz} - 2 \left( \frac{z^2}{r^2f^2}+\frac{x^2+y^2}{r^2} \right) \right)$$ Note if this equation can't be normalized another equation with similar proprieties would also be ...
2
votes
0answers
51 views

Unknown result in probability theory relating CDF of any density to the CDF of normal distribution

There is apparently a result in probability theory saying: If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = ...
2
votes
0answers
60 views

Gaussian Bayesian filtering with bound observation ($b_1<x<b_2$)

Suppose we have a Normal r.v $$ x \sim \mathcal{N}(\mu, \sigma^2) $$ and a Normal prior of $\mu$ $$ \mu \sim \mathcal{N}(\theta, \delta^2) $$ I know how to do the Bayesian update with a ...
2
votes
0answers
63 views

Probability estimation of a distances between samples and references - a classification problem

Background I am doing face recognition with an algorithm that is comparing a given test face to all other faces in a multidimensional space (face space). Essentially, this means, that the test face ...
2
votes
0answers
168 views

Proof of a gaussian integral turning into a cosine

I have a numerical evidence of $$\int_0^{1/2} \frac{1}{\sqrt{2\pi}\sigma_0x}\exp\left(-\frac{(\mu_0x-y)^2}{2\sigma_0^2x^2}\right)dx \approx 1+\cos(2\pi y),$$ where ...
2
votes
0answers
113 views

distribution of block occurrence of random vector in $\mathbb{Z}_2^n$

Given natural numbers $m, n \geq 2$ and a random vector $\mathbf{r}= (a_1,a_2,\cdots,a_n)\in\mathbb{Z}_2^n$. We define the $m$-circulant of $\mathbf{r}$ by the vector ...
2
votes
0answers
213 views

Convergence rate for the p.d.f. of a normalized mean to Gaussian (i..e Berry-Esseen for pdfs)

Berry-Esseen Theorem states that the rate of convergence of the probability distribution of normalized sample mean converges to Gaussian at rate $O(1/\sqrt{n})$ (given that certain conditions are met, ...
1
vote
0answers
6 views

References to papers/books that uses a kernel to smooth a discrete distribution

Since a kernel, such as Gaussian, is often used to smooth out the distribution of discrete points in 1D, 2D or 3D, I believe there must be some study materials or research work that have used this, ...
1
vote
0answers
8 views

Comparing normal distributions using a two sample Kolmogorov-Smirnov test

I have used a two sample Kolmogorov-Smirnov test to compare the distributions of two sets of data. I know that the K-S test is a non parametric test, however the distributions of data I'm comparing ...
1
vote
0answers
58 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
1
vote
0answers
8 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
1
vote
0answers
65 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
1
vote
0answers
17 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
1
vote
0answers
31 views

How to form Joint Probability Density from two Gaussian Distributions?

I've been reading the following paper entitled "An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning": http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3812643/ In Part ...
1
vote
0answers
28 views

Can this be solved analytically?

I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is $f(x) = 2 C_1 (X_1 - x) ...
1
vote
0answers
55 views

Conditional density based on 2 gaussian measurements

However intuitive, I don't understand the formulas for the conditional mean and variance from 2 gaussian measurements. I have not found anything relevant mainly because I don't think I'm searching ...
1
vote
0answers
44 views

Bivariate normal distribution; rotation; diagonal covariance matrix

Let $Z\sim N(0,\Sigma)$ with $$ \Sigma=\begin{pmatrix}\sigma_1^2 & p\sigma_1\sigma_2\\p\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix} $$ whereat $\sigma_i^2=\text{var}(Z_i), ...
1
vote
0answers
17 views

Binomial distribution vs Normal distribution

It is often said that the normal distribution "approximates" the binomial distribution. What is the precise mathematical expression of this fact?
1
vote
0answers
12 views

Expectation of product of Normal CDFs w.r.t. a bivariate Normal distribution?

I am trying to figure out if there is a closed form expression for the following expectation: $\int\int \phi(\gamma_1)\phi(\gamma_2) \mathcal{N}(\gamma\big|\mu, \Sigma)d\gamma_1 d\gamma_2$ where ...
1
vote
0answers
14 views

Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...