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Questions tagged [normal-distribution]

This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

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Distribution of two combined ML models

Due to the complexity of the problem, the problem was divided into two models: a stationary model and a model that corrects the stationary model for temporal effects, i.e. $X = X_{stat} + X_{time}$ ...
xbc68's user avatar
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Simple proof (avoiding Gamma function) for a Gaussian lower bound?

Let $g \sim N(0, I_n)$ be a standard multivariate Gaussian vector in $\mathbb{R}^n$. It can be shown via use of Gamma function identities and inequalities that $$ \sqrt{\frac{n}{n+1}} \leq \mathbb{E}\...
Drew Brady's user avatar
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posterior covariance of a gaussian process

I'm currently studying gaussian processes. In this framework we build "stochastic" functions f for instance $\mathbb{R}^N\mapsto\mathbb{R}$. If I've got $M$ input $X_i\in\mathbb{R}^N$ ...
Oersted's user avatar
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Uncertainties in measurements widen the normal distribution.

In redistricting (gerrymandering) analysis, a state is divided into districts where district vote shares are written in terms of one of two parties, so if district #1 has a vote share of 40%, it means ...
Ray J's user avatar
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1 answer
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Volume under a normal distribution in cartesian and polar not integrating to 1

This is my first calculus problem in several years as a professional. I'm checking to see that my integral of a normal distribution works, so that I can use it as a weighting function for ...
David Konyndyk's user avatar
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0 answers
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The moment of multivariate normal distribution

This is a computational problem I ran into while reading an article. I describe my question below: Let $\boldsymbol{Z}\sim N(0,I_{p\times p})$ and $\boldsymbol{y}_{i}\in \mathbb{R}^{p}$. We need to ...
Lop's user avatar
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Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
ChemGuy's user avatar
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An integral of several Gaussian densities

Let $f(x \mid \mu, \sigma)$ be the PDF of the Normal/Gaussian distribution. Is there a way to compute: $$ \int_{-\infty}^{\infty} \frac{f(x \mid \mu, \sigma_1)}{f(x \mid 0, \sigma_1)+f(x \mid 0, \...
Aleksandar Bojchevski's user avatar
4 votes
1 answer
67 views

Making a mixture of Gaussians unidentifiable?

Suppose that $A$ is a random variable such that $P(A=\sigma_{1})=P(A=\sigma_{2})=0.5$ for some $0<\sigma_{1}<\sigma_{2}$, and $$X\mid A\sim\mathcal{N}(0,A).$$ In words, the distribution of $X$ ...
Guy Ohayon's user avatar
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Maximum of $M$ random variables which are the maximum of $m$ normal distributed variables.

Let $X_{i,1}, \dots, X_{i, m}$ be a collection of normally distributed random variables $\mathcal{N}(0,1)$, and let $X_{i, (m)} = \max_{j\leq m}X_{i, j}$. I know from extreme value theory that $X_{i, ...
Faber's user avatar
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How can I solve this gaussian integral analytically? [closed]

This problem I am getting for my work. A, B, C are some constants. When the integration range is finite how can I solve this? I am having a problem with this integration $$\int_{\omega_a}^{\omega_b} \...
INDRANIL MAITI's user avatar
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Calculate optimal spacing for magnetic field measurement using Gaussian Multivariate likelihood distribution

I posted this question on Physics exchange as well, but is rather mathematic :) I have a vertical magnetometer configuration, and measure lines on the ground. I want to calculate the optimal spacing, ...
user387449's user avatar
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2 answers
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I want to check the positive definiteness of the matrix $\Lambda$

Background We consider independent and identically distributed (i.i.d.) random variables $$X_1,\ldots, X_n \overset{\text{i.i.d.}}{\sim} N_p(0, \Sigma).$$ Setup Let $\Sigma$ be a $p$th order positive ...
ytnb's user avatar
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Lower bound for variance of sum of with functions containing gaussian random vectors.

in my current research i am stuck with the following problem and search for nice lower bounds: Assume that the n-dimensional random $X_j$ is normal distributed, i.e. $X_j \sim N(0,\Sigma_j)$ where $\...
statuser123's user avatar
2 votes
1 answer
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How to show Geometric Brownian motion is not a Gaussian process?

Let's consider Geometric Brownian motion: $$ X_t = e^{\mu t + \sigma B_t} $$ where $B_t$ is Brownian motion. Question: How to prove that this process is not Gaussian? I understand that $B_t$ itself is ...
poiug07's user avatar
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Prove $\lim_{c\to\infty} P(X-\varepsilon<Y<X+\varepsilon\mid X>c,Y>c)=1$ for all $\varepsilon>0$ if $X,Y$ are i.i.d. Normal$(0,1)$?

Suppose random variables $X$ and $Y$ are i.i.d. Normal$(0,1)$. Consider the following events, where $\varepsilon>0, c>0$: $$\begin{align*} Q&=\{(x,y)\in\Bbb R^2: x>c, y>c\}\\ C&=\{(...
r.e.s.'s user avatar
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Generating MA Process in matlab using gaussian nosie

I have an MA process that is: $x[n] = a * w[n] + b * w[n - 1] + c * w[n - 2]$ Where $a, b, c \in \mathbb{R}$ and $w \sim \mathcal{N}(0, \sigma^{2})$ How can I generate $x[n]$ in Matlab ? I know that ...
Daniel Cohen's user avatar
4 votes
1 answer
83 views

Product of 2 normal variables with positive means

Suppose $X \sim N(\mu,1)$, $Y \sim N(\mu,1)$ are iid normal random variables with $\mu>0$. My research problem is finding out the asymptotics of the tail function of XY (since the explicit formula ...
BigFun's user avatar
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1 answer
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Calculating some Gaussian ratios

Let $N \geq 1$ be a positive integer, and let $w = (w_1, \dots, w_N)$ denote a positive sequence of real numbers. Let $\{g_n\}_{n \leq N}$ denote a sequence of iid standard Normal random variables. ...
Drew Brady's user avatar
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1 answer
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Are they sufficient conditions for joint normal distribution?

Suppose $\theta$, $u_1$, $u_2$ all follow normal distributions. Let $x_1$=$\theta+u_1$ and $x_2$=$\theta+u_2$. Are the above conditions sufficient to imply that $\theta$, $x_1$, and $x_2$ follow a ...
Ypbor's user avatar
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The Gaussian distribution of least area that contains an arbitrary number of distinct specified points in {$(x,y)\in\mathbb{R^2}:y>0$}

Definitions: $X=\bigcup_{i=1}^{3}(x_i,y_i)$ is a collection of $3$ distinct points in {$(x,y)\in\mathbb{R^2}:y>0$} $C\subset${$(x,y)\in\mathbb{R^2}:y>0$} is a bell curve symmetric about $x=x_0$ ...
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Is joint normal distribution flexible in correlation?

Suppose random variables $X$, $Y$, and $Z$ follow joint normal distribution. Conditional on $X$, we can calculate the correlation coefficient of $Y$ and $Z$. Is it always the case that this ...
Ypbor's user avatar
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0 answers
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Convolution of slightly multivariate Gaussians slightly modified

Starting with $ p(a) = \int p(a|b) p(b) db$ replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$ where $\tilde{D}$ is an additive diagonal covariance. Assuming ...
scleronomic's user avatar
2 votes
0 answers
45 views

Conditional Expectation for a multivariate normal distribution

I have $n$ random variables, $X_1, ..., X_n$, with covariance matrix, $\mathbf{\Sigma}^{n\times n}$ and I am trying to calculate $\mathbb{E}(X_1 | X_2 = x_1, ..., X_n = x_n)$. I am aware of the ...
Xerium's user avatar
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Reference for a normal CDF/PDF inequality

I am looking for references or any paper/book/report where the following inequality involving the standard Gaussian CDF and PDF appears or was used: $$ \Phi(x)\big(1-\Phi(x)\big) > \phi^2(x), \...
alexYYC2021's user avatar
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1 answer
42 views

The earnings of an employee follows the normal distribution $N(3000, 500)$. What is the probability that they earn more than $10 000$ in $3$ months?

Question : The monthly earnings of an employee follows the normal distribution $N(3000, 500)$. What is the probability that this employee earns more than $10 000$ in $3$ months? I've been thinking ...
Stephen's user avatar
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1 answer
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We throw a die 900 times. What is the probability that the frequency of occurrence of an even number is between $0.50$ and $0.55$?

Question : We throw a die $n=900$ times. What is the probability that the frequency of occurrence of an even number is between $0.50$ and $0.55$? Attempt: I think I have to use a normal distribution ...
Stephen's user avatar
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1 answer
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Derivatives of multivariate Gaussian

I do not understand how to take the second derivative of the Gaussian: While I am certain that the result is correct, I do not understand how to get from the first to the second line of the second-...
Make42's user avatar
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2 votes
1 answer
58 views

recusion formula for $X\sim \mathcal{N}(0,1)$

$X\sim \mathcal{N}(0,1)$ show that the moments of $X$ follow the follwing recusion formula $\mathbb{E}\left[ X^{n+1} \right]=n\mathbb{E}\left[ X^{n-1} \right] , n\ge 2$ the first induction step is ...
tom31415's user avatar
1 vote
0 answers
42 views

Sum of a Normal and a Chi2 distribution

I am doing some Monte Carlo exercises, and I have to derive the distribution of the following equation: $y= x + x \times u$ where $u\sim T_k$ is a Student $t$ with $k$ d.f. and $x\sim \sqrt{\frac{\chi^...
andrea's user avatar
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0 answers
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Help Understanding Theorem 2.2 Proof from Rue's Gaussian Markov Random Field

I am currently reading the book Gaussian Markov Random Fields by Rue (2005) and I am having trouble understanding the proof of Theorem 2.2. The theorem and its proof are as follows: Theorem 2.2: Let $...
clementine1001's user avatar
2 votes
2 answers
61 views

Conditional Expectation of dependent normal distribution

Suppose we have $Z_0, Z_1, Z_2$ all standard normal distributed and independent. And $ X = c+aZ_0 + aZ_1$ and $Y=c+aZ_0+aZ_2$ for $a,c \ge 0$. Is there a way to calculate $E[Y|X]$ only using ...
user007's user avatar
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2 votes
0 answers
43 views

How to force a product of two i.i.d. random variable to be gaussian

This question is related to this other question of mine: I realized that my original question was maybe too abitious, and I would like to discuss a much more limited version of it. Consider two real ...
Noumeno's user avatar
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1 vote
0 answers
30 views

Gaussian as mixture with uniform weights

Suppose $W$ is $\operatorname{Uniform}[0,1]$ distributed. Suppose $Y|W$ has known mean $\mu(W)$ and variance $\sigma^2(W)$, where $\mu$ and $\sigma$ satisfies some regularity conditions: (1) $\mu$ and ...
yrq's user avatar
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1 vote
0 answers
20 views

Expectation of log determinant of "weighted" Wishart matrix

Let $g_1,\dots,g_d$ be $n$ dimensional Gaussian iid vectors drawn from $N(0,I_n),$ and $p_1,\dots, p_d \in [0,1]$ be a discrete probability vector $\sum_i^d p_i=1,$ and define weighted Wishart matrix $...
kvphxga's user avatar
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1 answer
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How to derive resulting PDF of a random variable transformation involving discrete and continuous random variables?

This is a communication theory question, an example from the book Foundations of MIMO Communications. Suppose a symbol $s$ is drawn from a BPSK constellation (the book says distribution, but I do not ...
Userhanu's user avatar
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0 answers
13 views

Normal approximation to Poisson weighted sum

I am trying to come up with a way to approximate the sum given below by a normal distribution. $$ P(F| \overline{n}) = \sum_{n=0}^{\infty}P(F|n)P(n|\overline{n}) $$ F is the sum of IID random ...
John Smith's user avatar
0 votes
1 answer
24 views

Find $P(B_2 > 0, B_8 > 0)$

We need to compute $P(B_2 > 0, B_8 > 0)$ where $B_t$ is brownian motion Now, $P(B_2 > 0, B_8 > 0) = P(B_2 > 0, B_8 - B_2 > -B_2) = P(Z_1 > 0, Z_2 > -Z_1)$ where $Z_1 \sim N(0, ...
Harsh's user avatar
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0 answers
13 views

How to pull predictions from a set of overlapping normal curves

I have a problem writing a program that is beyond my stats level that I need help with. I have a set of pupil heights in a school, divided into 4 sets of class data. Each set of class heights can be ...
Steve Gilchrist's user avatar
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0 answers
19 views

Expected value of rectified linear function over white noise

I wondered if anyone could handle a rectified linear function's expectation. Looking online, I found this solution, which I assume is correct for one-dimensional $x$. $$ \begin{aligned} & \int_0^{\...
Nosrat Mohammadi's user avatar
1 vote
1 answer
75 views

Does sign of random noise really matter?

I was studying about Diffusion models in deep learning and came across the substitution $$x_t = \sqrt{\overline\alpha_t}x_0 + \sqrt{1-\overline\alpha_t}\epsilon\implies x_0=\dfrac{x_t-\sqrt{1-\...
insipidintegrator's user avatar
3 votes
1 answer
32 views

Expected value of this distribution

Let $a>0$ and let $X$ be a random variable with pdf $f(x)=\begin{cases}2\phi_\sigma(x)+\frac{\theta}{a}&\text{if $0\leq x \leq a$}\\0&\text{otherwise}\end{cases}$, where $\phi_\sigma(x)$ ...
BadIdeaException's user avatar
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0 answers
16 views

Model probabilistic time evolution from vector field distribution

Given an axis-aligned grid${}^1$ $x_1 , \ldots , x_n \in \mathbb{R}^2$. Let $\mu \in \mathbb{R}^{2n}$ and $\Sigma \in \mathbb{R}^{2n\times 2n}$ be the means and covariance matrix and assume we have a ...
Targon's user avatar
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2 votes
2 answers
53 views

Number of swaps needed in a random 8-ball pool rack to create a valid rack

I'm an engineer, not a mathematician, please don't judge me! I play a lot of pool and I like efficiency so I started wondering what the maximum number of ball swaps were needed to transform a random ...
Jason Stonier's user avatar
1 vote
1 answer
28 views

Using a double integral to find the CDF of the standard Cauchy distribution

This question overlaps with the second question here. Blitzstein and Hwang's "Introduction to probability" says: "Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $T = X / Y$." ...
johnsmith's user avatar
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0 answers
26 views

Formula for jointly correlated variables

I've always seen the following expression for normal correlated variables which is considered quite basic: \begin{equation} A=\rho Z+\sqrt{1-\rho^2}\epsilon \end{equation} I understand that it follows ...
vsa's user avatar
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0 votes
0 answers
26 views

expectation of logarithm of scaled chi-squared random variable

Suppose $x_1,\dots, x_n\sim N(0,1)$ are iid Gaussians, and $\lambda_1,\dots,\lambda_n > 0$ are positive scalars. Define scaled chi-squared random variable $Y = \sum_i^n \lambda_i x_i^2,$ where we ...
kvphxga's user avatar
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0 votes
1 answer
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If $X\sim\mathcal N(x,\Sigma)$, what is $\operatorname E\left[\left\|\Sigma^{-1}(X-x)\right\|^2\right]$?

The question is in the title. If $\Sigma=\sigma^2I_d$, then we easily calculate $$\operatorname E\left[\left\|\Sigma^{-1}(X-x)\right\|^2\right]=\frac1{\sigma^2}\tag1.$$ However, the general case seems ...
0xbadf00d's user avatar
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1 vote
0 answers
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Recurrence relation for orthogonal polynomials with power law weight function

The objective is to find the recurrence relation for orthogonal polynomials with respect to the scalar product: $$\langle f,g \rangle = \int_a^b f(x) g(x) w(x) dx,$$ where $0<a<b<\infty$, $w(...
gb2718's user avatar
  • 21
1 vote
1 answer
52 views

Estimate the correlation coefficient of a two-dimentional normal distribution $(X,Y)$, given some samples of $(|X|, |Y|)$

I have two random variables $X,Y$, whose joint distribution is a two-dimensional normal distribution, and the expectations of both $X,Y$ are zero. Let $\rho={\rm cov}(X,Y)$ be their correlation ...
zemora's user avatar
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