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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39 views

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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24 views

Quotient of two Gaussian densities

The matrix cookbook contains formulas for the product of two multivariate Gaussians, but doesn't appear to contain formulas for the quotient of two Gaussians. $$ \frac{\mathcal{N}(\mathbf{m}_1, \...
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36 views

Existence of solution of equation involving normal distribution

I've tried to show that the following equation has a solution: \begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ ...
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47 views

Weak convergence of Poisson distributed random variables

I am stucked in the middle of an exercise: Let $$X_n,Y_m$$ independent random variables having the Poisson distribution with parameters n and m respectively. Show that $$\frac{(X_n-n)-(Y_m-m)}{\sqrt{...
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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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47 views

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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16 views

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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19 views

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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38 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi \...
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39 views

Product of matrix-valued normal densities and Kronecker product

I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions. Here is what I've tried in a special case I ...
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18 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} Y_1 ...
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Hypothesis test in Bayesian statistics

Let $X\sim N(\theta,1)$ and 5 independent observations $X=(4.9,5.6,5.1,4.6,3.6)$. The prior probability that $\theta=4.01$ is $0.5$. The remain values of $\theta$ are given the density of $g(\...
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38 views

Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: $...
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21 views

Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but $$f(\...
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58 views

Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
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38 views

What is the distribution of $\frac{\sum(x_t-a_t)^2}{\sum(x_t-b_t)^2}$

Let $x_t, t \geq 1$, be a sequence of independent random variables, $x_t \sim N(a_t,\sigma^2), t \geq 1$, $a_t, b_t \in \mathbb{R}$. What is the distribution of $S_n$, where: $$S_n=\frac{\sum_{t=1}^n(...
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32 views

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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96 views

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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48 views

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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41 views

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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43 views

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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24 views

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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37 views

Convergence of normal distributed random variables

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables with the property that $N(m_n, \sigma_n^2)$. Prove that if $X_n$ converges in distribution to $X$ then $X \sim N(m,\sigma^2)$ where $m =...
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A closed-form formula for Cov(X,Y) when X and Y are normal random vectors?

I cannot figure out one step given in my textbook, [1] Mixed Models. Theory and Applications by E. Demidenko. I study the Linear Mixed Effects (LME) model in the following form: $$\mathbf{y_i}=\...
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Distribution of $X_3$ given that $X_1 + X_2 =1$ for $(X_1,X_2,X_3)$ centered gaussian with given covariance matrix

Let $X=(X_1,X_2,X_3)$ have a multivariate normal distribution with $EX_1 = EX_2 = EX_3 = 0$ and covariance matrix: $ \left( \begin{array}{ccc} 2 & -1 & 1 \\ -1 & 5 & 0 \\ 1 & ...
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25 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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29 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 ...
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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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49 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 $$ E(X^k~|~X>1/\sqrt{d^{1-\...
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Can you perform a two-sample t-test with a moderately skewed sample distribution

This was a question on my AP Stat test that I got wrong and I am not sure if it is correct: Essentially the question stated you have two samples both of which don't satisfy the large counts condition ...
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25 views

Is the following function symmetric?

I am reading the paper Hierarchical Clustering of a Mixture Model by Goldberger et al. On the page 2 of this paper they define the following function: $ d\big(\mathcal N(\mathbb\mu_1, \Sigma_1),\;\...
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74 views

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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41 views

expectation of normal-wishart distribution

I want to compute $ E[\mu\Lambda] $ for a normal-wishart distribution how can i compute it? A normal-wishart distribution is defined as below: $$ (\mu,\Lambda)=NW(\mu,\Lambda|\mu_0,\lambda,W,v)=N(\mu|...
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23 views

Finding conditional probability distribution (X|Y) from (Y|X)

$(Y,X)$ has a joint distribution where the marginal of $X$ is a standard normal and $Y|X \sim U \left[|X|-\frac{1}{2},|X|+\frac{1}{2}\right] $ where $U[a,b] $ means uniform in the interval [a,b]. How ...
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Bivariate normal: normal difference distribution

I know that the difference of two multivariate or univariate normally distributed random variables produces another (multivariate) normal random variable. But, what happens when you take the ...
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54 views

Generating random numbers of bell curve distribution

I want to generate random numbers that fit a bell curve distribution. Basicly, I need random numbers from 0 to 1, but I wish to have a high likelihood of it being close to 0.5, but not guaranteed, ...
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Find likely maximum distance from center of gaussian sphere in high dimensions

I am testing a clustering algorithm in high dimensions. I want to see how it behaves as I allow the clusters to get closer and closer, but it must work perfectly for "well separated" clusters. I need ...
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26 views

Probability distribution of $g^h f f^h g$

We define an $k \times k$ complex matrix $M=[V \, \mathbf{0}]$, where matrix $V$ is $k \times (k-l)$ dimensional and is unitary, and $\mathbf{0}$ is the $k \times l$ zero matrix. Let vector $f$ be a $...
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26 views

Converting log-scaled volume density to number fraction

I have a log scaled volume density distribution, $q_{3,log}$ from which I want to get number fraction, $\Delta Q_0$ with normal scale. So to transform $q_{3,log}$ to $\Delta Q_3$ the used relation is $...
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Roots of an equation with normally distributed variable

Consider the following equation: $p\left(1-\int _{\mu}^{x} f(y)dy\right) \left[p\left(1-\int _{\mu}^{x} f(y)dy\right)+(1-p)q \right]-xf(x)p(1-p)q=0$, where $p,q \in [0,1]$, $f(\cdot)$ is the ...
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340 views

Laplace transform of normal distribution function?

In my notes this was left as an exercise and I am a bit rusty with my calculus. Starting with the definitions: $$\mathcal{L}_X(t) = \mathbb{E}[e^{-tX}] = \int_0^\infty e^{-Xt}f(t)dt \;\;\text{ and }\;\...
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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{...
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Show that this conversion between $\Phi$ and erf(z) holds for all z

I am trying to wrap my head around the connections between the standard Normal distribution and the error function. I could use some help working through the following problem. Show that the ...
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Problem on Bivariate normal distribution

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + ...
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Finding the marginal distributions of a binormal random variable

Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where, $\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m = \...