# Tagged Questions

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

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### Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
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### Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
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### Problem on Mann Whitney U test statistic

Let $X_1, X_2, \ldots, X_m \sim N(\mu_1, \Sigma)$ and $Y_1, Y_2, \ldots, Y_n \sim N(\mu_2, \Sigma)$. (Here, $\Sigma$ is the variance-covariance matrix of the 2 multivariate Normal distributions ...
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### Normal Distrubition Question - How many wires will meet specifications?

Wires manufactured for use in a certain computer system are specified to have resistances between 0.12 ohm and 0.14 ohm, the actual measured resistances of the wires produced by company A have a ...
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### Integrating a Random Variable and establishing the maximum of a related function

Frequency Regulation of a Power Grid I have a battery that is used to regulate the frequency of a power grid. That is, as the grid frequency varies about it’s ideal value, $f_{nom}$, the battery ...
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### Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
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### Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
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### Variance of truncated multivariate Gaussian

Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets. Let $P_S$ be the ...
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### Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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### Inequality involving the sum of normal random variables

Problem: Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables from the normal distribution with mean equal to 1.5 and standard deviation equal to 4. Show that with ...
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### Bivariate normal distribution hazard rate

Suppose $(X,Y)$ is bivariate normal with $\mu=\begin{pmatrix} 0 \\ 0 \end{pmatrix}$, $\Sigma=\begin{pmatrix} \sigma^2 & \rho \sigma^2 \\ \rho \sigma^2 & \sigma^2 \end{pmatrix}$ Is it ...
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### If $B_t - B_s, \ 0\leq s < t,$ is normally distributed, there are constants $C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n$

I am working on the following problem: Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive ...
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### What is the expected distance from the mean of a multivariate Gaussian?

For a multivariate Gaussian distribution $p(x) = N(x\mid \mu,\Sigma)$, what is $E[\|x-\mu\|]$? I know from this question that $E[|x-\mu|]=\sigma\sqrt{2/\pi}$ for univariate Gaussians. But I couldn'...
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### Multivariate distribution with the same kurtosis as normal distribution

Good morning. I am writing a thesis about testing multivariate normality. I would like to do a comparison of power of some tests against given alternatives based on Monte Carlo simulations. I have a ...
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### How to do the convolution of a normal distribution with a truncated exponential distribution?

I have a random variable $A$ with $A = B + C$, where $B$ is a normal distribution with the usual range $(-\infty , +\infty)$ and $C$ is a truncated exponential distribution of range $(a,b)$. How to ...
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### Distribution of squared multivariate normal random variable

Let $W\sim MVN(\mu, \Sigma)$, here $W$ and $\mu$ are $k\times 1$ vector and $\Sigma$ is $k\times k$ symmetric matrix. And the diagonal elements of $\Sigma$ are equal to one. For this multivariate ...
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### Theoretical distribution of a random variable

Martin has $n$ words, and he wants to make a computer program that chooses for him $k$ words (and shows them to him), where $k \le n$, for as many times as he clicks a button until all of the words ...
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### What is the Edgeworth Expansion of the binomial distribution?

For a standardized binomial distributed random variable $\tilde B_n$ we have P(\tilde B_n\le x) = \Phi(x) + \frac {q-p}{6\sqrt{npq}} (1-x^2) \phi(x) + \frac{R_1\left(np+x\sqrt{npq}\right)}{\sqrt{...