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

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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: ...
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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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91 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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27 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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18 views

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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38 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 ...
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12 views

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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38 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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23 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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34 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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28 views

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: ...
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28 views

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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26 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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46 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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48 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 $$ ...
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12 views

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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23 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, ...
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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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34 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: $$ ...
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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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20 views

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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34 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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22 views

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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25 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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17 views

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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39 views

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) + ...
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17 views

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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31 views

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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19 views

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 ...
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22 views

Normal $(\frac{n-1}{n})$-percentile asymptotic to $(2\log n)^{1/2}$?

I am working from Durrett's Probability: Theory and Examples, and I have encountered the following question: Suppose that $X$ is normally distributed, and $b_n$ is defined by ...
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31 views

Normal Distribution with equal probability P(x | y)

Hi I am solving one problem based Bayes' formula. I need to calculate the normal distribution of P(x|y). The following data is given. P(x | y = 0) = N(x1,0,1) and P(x | y = 1) = N(x2,0,16) where N ...
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22 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
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40 views

Joint and Conditional pdf of Two Normal r.v.'s

Given two uncorrelated Gaussian random variables, X ~ $\mathcal{N}(0,1)$ Y ~ $N(0,1)$ Find $f_{y|x}(y|x)$, $f_{x,y}(x,y)$ If I can find either the conditional probability or the joint ...
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28 views

Does the standard deviation of a normal distribution change when looking at a shorter period?

Say we are given a normal distribution with mean $\mu$ and standard deviation $\sigma$, where $\mu$ is the expectation in time period $T$. Now, say you need to use the distribution over a shorter ...
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19 views

Distribution of ratio of 2 points drawn from normal distribution?

Let's say we have a known normal distribution $N(\mu,\sigma^2)$. I now draw 2 points $p1$ and $p2$ randomly from this Gaussian distribution for every observation, and repeat this process large number ...
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22 views

Can't figure out problem using joint density with bivariate normal distribution.

Not even sure where to start with part (b) for the problem below. For part (a) assuming the worker knows her own skill level and the prevailing wage, I got: y1 > y 0, or S1 - S0 > ln(w0/w1) for ...
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17 views

What is the expected number of extreme points in a set of points drawn from a normal distribution?

Say I construct a set of points $S$ by drawing $k$ points from $\mathbb{R}^{n}$ independently from say a gaussian distribution*. I am wondering how the expected size of the set $E$ consisting of the ...
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41 views

Cumulative distribution function of function of normal random variables

I have $X_1 \sim N(0, 4), X_2 \sim N(0, 4), X_3 \sim N(3, 1), X_4 \sim N(1, 9)$, they are independent. I need to find cumulative distribution function of $\xi = \frac{X_3X_2 + X_4X_1}{ \sqrt{X_4^2 + ...
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a puzzling result for quadratic ratio of normals

Note: Below I use $\approx$ to indicate equivalence in distributions and use $\sim$ to indicate that how the distribution law of a certain variable is defined. Let $X_{1,i}$ and $X_{2,i}$ be two ...
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Bayesian Gaussian Mixture model

I am trying to fit basic Gaussian mixture with a Bayesian model. My likelihood function is Gaussian, with std=1, and the only parameter is the mean, chosen from $\{0,1,\dots,14,15\}$ and my prior is ...
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10 views

Multi sample chi-square distribution

I have a question from my assignment. For a total of 150 households selected randomly from 3 cities -- A, B and C -- of 50 each, the sample mean and covariance matrices are as below: For A, ...
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40 views

The distribution of gaussian 2D vector given the distribution of average

My question: Let $X_1, X_2\sim N(\theta,1)$. Let $\bar X = aX_1+(1-a)X_2$ for $0<a<1$. Find the distribution of $(X_1,X_2)$ given $\bar X$. Update: My previous try is wrong. I delete it. ...
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28 views

Distribution of conditional expected value

Motivated from an application in economics, I would be interested in a simplified solution for $C$, where \begin{equation*} C = E\bigg[exp \bigg\{-\frac{1}{2} \frac{(E[\theta|S]-a)^2}{b}\bigg\}\bigg] ...
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57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
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35 views

How to sample multivariate random normals?

Suppose $\Omega=(0,1)$, $\mathcal{B}$ is the Borel $\sigma$-algebra on $(0,1)$ and $P$ the Lebesgue measure. Let $\Phi:\mathbb{R}\to(0,1)$ denote the standard normal CDF. Then, we can generate $X\sim ...
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Covariance estimation and Graphical Modelling

I've started reading on Convariance Matrix estimation through Graphical model in high-dimensional situation. But I have several questions. Suppose, $X_i \overset{iid}{\sim} N_p(\mu,\Sigma)$, ...
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31 views

Characteristic function of $\chi^2$ distribution with one degree of freedom

Just for my own curiosity, I'm trying to derive the characteristic function of $X\sim\chi^2_1$, the $\chi^2$ distribution with 1 degree of freedom. According to wikipedia, it is $$ ...
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41 views

Prove that $\Bbb P[\max_{k\le n}x_k<(2+)\sqrt{\log n}$ as $n\to \infty]=1$?

By the Borel-Cantelli lemma to prove that $\Bbb P[\max_{k\le n}x_k<(2+)\sqrt{\log n}$ as $n\to \infty]=1$ with a number $(2+)$ a shade more than 2. The hint is to use $(1-x)^n>1-nx$. ...