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

learn more… | top users | synonyms

6
votes
0answers
277 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
79 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
1k 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 ...
4
votes
0answers
136 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
20 views

$L^2$ limit of Gaussian random variables

Let $X_n$ be a sequence of Gaussian random variables defined on the same probability space. The statement is that if $X_n$ converges to some random variable $X$ in $L^2$-sense, then $X$ is also ...
3
votes
0answers
97 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
63 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
28 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
94 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\\ ...
3
votes
0answers
106 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 ...
3
votes
0answers
96 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
79 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
148 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
67 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? ...
2
votes
0answers
29 views

How to calculate this kind of probability for a normal distribution?

here is my question. I have a normal distribution with known mean and variance. Say the mean is 3 and the Var. is 2. what is the probability that the random variable is taking value 2.9? If I plug ...
2
votes
0answers
101 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = ...
2
votes
0answers
33 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...
2
votes
0answers
45 views

Finite discrete approximation to the normal distribution

I wish to derive a finite (that is, which has a finite support) discrete approximation to a normal distribution, with the following considerations: It should have exactly the same mean and variance ...
2
votes
0answers
149 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
59 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
35 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
56 views

Integral with truncated normal distribution

I am attempting to determine closed form equations for several integrals. Suppose $X=N(\mu,\sigma)$ is normally distributed with PDF $f(x)$ and CDF $F(x)$. $$\int_{T}^{\infty} xf(x)dx $$ ...
2
votes
0answers
26 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
58 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
110 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
93 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
40 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
34 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
163 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
366 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
35 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
80 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
998 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
111 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
55 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
65 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
65 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
173 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
114 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
239 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
16 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
1
vote
0answers
22 views

how to prove $\mathop {\lim }\limits_{n \to \infty } {\{\Phi [(1 - \varepsilon )\sqrt {2\log n} ]\}^n}=0$?

$\Phi (x)$ is the distribution function of standard normal distribution. $\varepsilon$ is some positive tiny number that is less than 1. How to prove this beautiful and important limitation: ...
1
vote
0answers
16 views

Explanation of Approximation for Integral Over Gaussian Distribution

I am reading an optics textbook that uses the following integral to evaluate the power squared in the lower tail of the following Gaussian integral. $$\frac{1}{{{\sigma _P} \cdot \sqrt {2 \cdot \pi } ...
1
vote
0answers
18 views

Empirical Rule. Is it applicable in this case?

So I ran in this problem: I have to test whether Empirical Rule is applicable. Proportions I got is 73%, 94,7% and 99.1% (within one, two and three standard deviations). I'm worried about 73%. This is ...
1
vote
0answers
25 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
1
vote
0answers
11 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
1
vote
0answers
34 views

A property of the hazard function of the normal distribution

I have a problem that I can't figure out. Define $$\Gamma\left(x\right):=\frac{\phi(x)}{1-\Phi(x)}$$ where $\phi(x)$, $\Phi(x)$ are the density respectively cumulative distribution function of the ...
1
vote
0answers
14 views

How to check $H_0$ hypothesis using Pearson's criteria?

How to check hypothesis by using Pearson's criteria ( $\chi^2$ test), that $H_0:$ random variable $X$ is normally distributed given that $k=7$ (count of intervals) and $\alpha=0.1 $ (significance ...
1
vote
0answers
20 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...