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

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1answer
33 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
-2
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0answers
11 views

Distribution of a sum of squared normal variables? [on hold]

What is the distribution of $\sum X_i^2$ ? Where $X_i\sim N(\mu_i,\sigma_i^2)$, for $i=1,..,n$. Thanks a lot for answers!
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0answers
6 views

Optimize polling frequency between producer and consumer to achieve minimum waiting time

Background: I am trying to optimize what we call AJAX request polling frequency in the domain of web design, and I wanted to check if I could use some help from math guys to explore a better ...
1
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1answer
14 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
8
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3answers
277 views
+50

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
2
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0answers
46 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
2
votes
1answer
14 views

Inverse of Gaussian CDF, Sum

Consider the following setting. Let $k = 1, \ldots, n$ and define $$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$ where $\Phi$ is the inverse of the CDF of a standard normal. I noticed numerically ...
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0answers
6 views

Bounding the ratio of sample covariance to population covariance

I am looking to bound the Kullback Leibler divergence of two mean centered Gaussian laws $\mathbb{P}_1=\mathcal{N}(0, \Sigma)$ and $\mathbb{P}_2=\mathcal{N}(0, \hat{\Sigma})$ where $\Sigma$ is the ...
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0answers
8 views

Separability Hypothesis Test

Let $M_1 \sim \mathcal{N}(\mu_1,\mu_1)$ and $M_2 \sim \mathcal(\mu_2,\mu_2)$ be normally distributed and independent random variables, each depending on only one parameter $\mu_n$ ($n = 1,2$). Let's ...
0
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0answers
14 views

integral of trace function

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
0
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1answer
1k views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
1
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0answers
21 views

Integer and fractional pars of Gaussian random variables

Are there any interesting results known about integer and fractional parts of $\xi \in \mathcal N(0, \sigma^2)$? In particular, I am interested in their expected values and covariance matrix.
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0answers
40 views

Distribution of Product of Matrices holding Multivariate Random Normal Variable Observations

We have matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ that hold observations from multivariate normal distributions. Each matrix represents $\boldsymbol{n}$ observations of $\boldsymbol{m}$ variables ...
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0answers
20 views

Joint distribution of two normal marginal distributions

My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence. Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the ...
1
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1answer
503 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
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1answer
35 views

Joint characteristic function of $x$ and $y=x^2$ if $x$ is the standard normal variable

How to find the joint characteristic function of $x$ and $y=x^2$ if $x$ is standard normal variable with mean $0$ and variance $1$?
2
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0answers
49 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
4
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1answer
93 views

PDF of $X = \max\{X_1,X_2\}$, being $X_1$ and $X_2$ independent Normal distributed random variables

Let $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, what's the CDF of $X = \max\{X_1,X_2\}$? Both variables are assumed to be independent. I tried the ...
0
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0answers
10 views

Estimation effectiveness of two normal-distributed variables

You have two processes of measuring the air pollution, $X$ and $Y$. Both processes deliver values which are normal distributed around $\mu$: $X ~ N(\mu, \sigma_x^2)$ and $Y ~ N(\mu, \sigma_y^2)$. I ...
0
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1answer
24 views

Understanding standardization for normal distribution

Let X be normally distributed random variable with expected value $\mu$ and standard deviation $\sigma$, then its СDF is: $$ F(x)=\frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x ...
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0answers
10 views

Correlate normal shocks

I am trying to generate some random standard normal variables and correlate them In particular I want: $$ \bf Y \sim \mathcal N(0, \Sigma) $$ where $\textbf{Y} = (Y_1,\dots,Y_n)$ is the vector I ...
1
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2answers
28 views

$\frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx$ - Normal Distribuition

I have read in one of my finance books (Asset Pricing - John H. Cochrane) that there is this identity: \begin{equation} \begin{split} \frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx ...
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1answer
548 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
0
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1answer
25 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
1
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1answer
583 views

The characteristic function of a multivariate normal distributed random variable

The characteristic function of a random variable $X$ is defined as $\hat{X}(\theta)=\mathbb{E}(e^{i\theta X})$. If $X$ is a normally distributed random variable with mean $\mu$ and standard deviation ...
0
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1answer
33 views

Black-Scholes SDE solution help

I am trying to solve the Black Scholes SDE, but got really stuck. I have done most of the derivation but the integral seem intractable to me. The integrals look bit like the Normal Distributions PDF, ...
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0answers
18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
2
votes
2answers
19 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
1
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1answer
419 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 ...
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0answers
21 views

Expected value log-normal variable

Suppose $X_t$ and $Y_t$ both have standard normal distributions with mean zero and variance 1. What is the $E_t[e^{aX_t + bY_t^2}]$ where $a$ and $b$ are constants? It should be something like: ...
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1answer
31 views

Prove inequality for tail of normal distribution

I have to prove this inequality for $x>0$. I have no idea how to even start this. I would appreciate any help. $$\frac{x^{2}}{1+x^{2}}\frac{1}{x}\exp \left( \frac{-x^{2}}{2}\right)\leqslant ...
-1
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0answers
9 views

Application of Jacobian matrix determinant in multivariate normal distribution transformation

Consider a multivariate normal distribution in variable x with mean μ and covariance Σ. Show that if we make the linear transformation $y = Ax + b$ then the transformed variable y is distributed ...
2
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1answer
47 views

Expectation of a function of a normally distributed random variable

Consider that I have to produce this result: $$E[u(W_0+r(\theta))] = u(W_0)+\theta-\frac 12\rho\sigma^2$$ From this: $$ E[u(W_0+r(\theta))] = \int_{-\infty}^\infty u(w_0+r) \frac{1}{\sigma ...
1
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1answer
2k views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
1
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2answers
25 views

Confidence Intervals

Let $x_1, \ldots, x_n$ be a sample from a normal population having unknown mean and variance. Let $\bar{x}$ be the average of the first n of them. What is the distribution of $x_{n+1} - \bar{x}$? If ...
0
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1answer
448 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
2
votes
1answer
391 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
1
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1answer
16 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
0
votes
2answers
20 views

convergence in distribution of truncated gaussian variables

Let $X$ be a random variable which is distributed normally with mean $\mu=0$ and variance $\sigma=1$. Suppose that $X_n$ is a random variable for any positive integer $n$ with truncated normal ...
2
votes
1answer
40 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose variance is distributed Log ...
1
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0answers
37 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 ...
0
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1answer
21 views

Find the value for z(0.1) from a distribution table?

I'm doing a statistics course, and I thought I had no problems using distribution tables to find values. For example, for the Gauss distribution, if I want $\Phi(-2)$, I will do $1 - \Phi(2)$ because ...
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0answers
14 views

segments of normal distribution are normally distributed?

I need a hint how to prove following: Log fold changes follow normal distribution. On the plot you can see log2 fold changes versus mean. If I segment log2 fold changes into the bins, so that I have ...
0
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1answer
18 views

How to find the PDF for Y=|X-1|?

Given $X\sim N(0,1)$ and $Y=|X-1|$. Find the PDF of $Y$. I tried to discussed when $x>1$ and $x\le1$, but this gives me two different functions and I have no idea how to combine them. However, ...
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0answers
47 views

The higher moments of truncated Gaussian

We assume that $$X \sim N(0,1/d),$$ where $d\rightarrow \infty$. For $\delta > 0$ sufficiently small. My question is, what is the correct order (in terms of $d$) of $$ ...
0
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0answers
20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
1
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0answers
17 views

Expectation of the matrix product $Q A (Q A)^h$

Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$. We define $A$ as an $n \times k$ ...
2
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0answers
34 views

Compound Distribution — Log Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (mean of the log of ...
3
votes
0answers
54 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
0
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0answers
24 views

Could you please explain how do we get the second term while computing $E[f(Y)]$

I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform ...