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

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20 views

Continuity Correction with replacement

An urn contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. What is the probability of drawing more than 75 red marbles? My attempt: $n=100, ...
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32 views

Average minimum distance

Let $\mathbf{u} =\begin{bmatrix}u_1 & u_2 & \dots & u_N \end{bmatrix}^T$ and $\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \dots & v_N\end{bmatrix}^T$. All the elements of ...
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43 views

Mean & SD of Sampling Distribution

A population consists of $4$ numbers $\{0, 2, 4, 6\}$. Consider drawing a random sample of size $n = 2$ with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal ...
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73 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
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38 views

Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous?

Let $X$ be a standard Gaussian random variable. Is the $\mathbb R^2$-valued random variable $(X,X)$ absolutely continuous ? I don't understand the question here. Now $X$ has density ...
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33 views

Conditional covariance in gaussian graphical models

I have a hypothesis, but I'm not sure if its true. The Wikipedia page states that if the covariance matrix is given by $$\Sigma=\left[\begin{matrix} A & B \\ B^T & C \end{matrix}\right]$$ ...
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249 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 ...
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69 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
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26 views

We said the data is normally distributed, based on the raw data or residual?

I have a confusing regarding the assumption test for the data, in some theory were said that there are three assumption of data as we called as "good" data: Independent Normally distributed ...
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18 views

The space of all normal covariances matrices

Let $\cal C$ be the space of all $k-$variate normal covariance matrices and $\cal M$ be the set of all $k\times k$ symmetric positive semi-definite matrices. As we know that if $k=1$ then ${\cal ...
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36 views

Frechet differentiability, asymptotic normality

I try to prove the asymptotic normality from the Frechet differentiability. Consider $$T(G)-T(F)=L_{F}(G-F)+o\left(d_{\star}(G,F)\right)$$ and ...
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20 views

Linear Gaussian system, covariance of the normalisation constant

If we have the following multivariate Gaussian distributions: $$p(x) = N(x|\mu_x,\Sigma_x)$$ $$p(y|x) = N(y|Ax + b, \Sigma_y)$$ Now how can you deduce p(y) ? p(y) is called the normalisation ...
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29 views

intergral of the product of 2 multivariate Gaussian distribution

Suppose there are the following relationships between $x,y,w$, $$\begin{align}p(x,y) &= N(\mu_1, \Sigma_1)\\ p(x\mid w) &= N(\mu_2,\Sigma_2)\end{align},$$ is it possible to compute $p(y\mid ...
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64 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
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23 views

estimate normal distribution parameters by $n$ largest samples

If I have the $n$ largest out of $m$ values of a sample from independent normal distributed random variables $\mathbb{X}_1,\dots,\mathbb{X}_m\sim\mathcal{N}(\mu,\sigma)$ with unknown parameters ...
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29 views

What is the probability the maximum sample value comes from one of two random distributions?

Let $X_1$ and $X_2$ be randomly distributed variables with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$. Samples of size of $n_1$ and $n_2$ are drawn from each ...
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61 views

conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
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22 views

proof of As ~ N(A$\mu$, A$\Sigma$A')

assume that s is a vector of states which is distributed according to a gaussian with mean $\mu$ and variance $\Sigma$. A is the state transition matrix How can I proof that As ~ N(A$\mu$, ...
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68 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
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105 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
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76 views

Interval of non-uniformly distributed set of numbers adjusted that it properly excludes extremes

Let's say I have an interval of numbers from 1 to 9 with the following frequency of distribution: numbers 1, 2 and 3 about 20 occurrences number 6 has 2 occurrences and number 9 has only ...
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146 views

Multivariate Distribution & Bayes Rule

Suppose I have that an unknown vector, x, where x is drawn from the following distribution$ \bigl(\begin{smallmatrix} x_1 \\ x_2 \end{smallmatrix} \bigr)$ ~ $N\bigl(0, \bigl[\begin{matrix} \sigma^2_1 ...
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517 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
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68 views

Conditional multivariate normal distribution

If $X = [X_1,\dots,X_n]$ is follows a multivariate normal distribution $\mathcal{N}(\mu,\Sigma)$, are there any (closed form) results known for the distribution of $[X_1,\dots,X_i \mid l_{i+1} < ...
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148 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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34 views

Finding the distribution under a new measure

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is ...
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34 views

Confusion related to gaussian

I have this confusion related to gaussian distribution Do we need to have something like $e^{-\frac{x^2}{2}}$ to be called gaussian or $e^{-{x^2}}$ is enough to be called Gaussian. I was reading this ...
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292 views

Almost sure convergence of maximum in a sequence of Gaussian random variables

Let $X_1, X_2,\ldots,X_n$ be an i.i.d. sequence of standard Gaussian variables and $M_n=\max(X_1, X_2,\ldots,X_n)$. I am trying to understand the mechanics of the proof of almost sure convergence ...
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28 views

Normal Distribution and test hypothesis

I have done 3 experiments. For each one of them, I have repeated the same experiment 100 times. Which gives me three sets of 100 numbers. Experiment 1: for number 30 ---> 100 results Experiment 2: for ...
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54 views

How to learn mixture Gaussian with inequality constraint of component variances

Let $f_1(x)$,…,$f_n(x)$ be Gaussian density functions with different parameters, $\mu_i$ and $\sigma_i$ are the parameters (mean and variance) of the Gaussian component i, and $w_1,\ldots,w_n$ be real ...
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61 views

$\frac{\partial}{\partial\theta}\phi'\mu+\frac{\alpha\phi'\Sigma\phi}{2}=0$

Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ...
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96 views

Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
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182 views

Berry-Esseen Theorem-like result with fourth central moment instead of third absolute moment

Let $X_i$, $i=1,\ldots,n$ be i.i.d. random variables with $E[X_i]=\mu$, $E[(X_i-\mu)^2]=\sigma^2$, and $E[(X_i-\mu)^4]=\kappa$. I am interested in approximating the distribution of ...
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31 views

Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
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1k 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 ...
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1k views

calculate probability without table

my question is related to normal distribution,namely as i know in GRE quantity section,there could be question related to normal distribution,but of course we will not have table,o how can we ...
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54 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
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129 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
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86 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
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93 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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143 views

Probability that a point from one normal distribution is greater than points taken from several other distributions?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that each is greater than all the others. For example: ...
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138 views

Problem involving the bivariate normal distribution

If $X$ and $Y$ have a bivariate normal distribution with $\mu(x)=\mu(y)=0$, $\rho=0$, $\sigma(x)=\sigma(y)=10$. Find the following: A) The probability of getting a point $(x,y)$ inside the ...
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218 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
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268 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
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41 views

algorithm to use to balance a set of IPs into a set of buckets

So we have a set of IP addresses (~3000) and want to balance them into 4 different buckets. What we are doing now is very simple by treating the last part of the IP as integer and mod it by 4. e.g. ...
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107 views

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation between the real return on the asset and the stock market return. Okay, trying to be more explicit: Let ...
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86 views

Product of 2 Gaussian Distributions with Different Variables

Sorry, I asked the original question improperly so I am rephrasing it. What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ and $f_{Y|X,PA}$ are both ...
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30 views

What is the product of the average x and the average (1/x), where x is normally distributed.

Has anyone ever seen a solution for the following...? $$ \left( \frac{1}{n}\sum_{i=1}^{n} x_{i} \right) \times \left( \frac{1}{n} \sum_{i=1}^{n} \frac{1}{x_{i}}\right) $$ where the x values are from ...
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259 views

Cramer’s decomposition theorem - find normal distributions within a normal distribution.

I know that Cramer's decomposition theorem says that any normal distribution can be expressed as the sum of multiple normal distributions. I have been searching for a method to divide a data set that ...
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98 views

continued fraction multivariate normal distribution?

After searching for a while, I wonder if a continued fraction representation exists for the multivariate Mills ratio $P(Z \geq x)/\phi_Z(x)$ where $Z \tilde\, N(\mu,\Sigma)$. The representation ...