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

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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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20 views

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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26 views

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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39 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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26 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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56 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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22 views

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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27 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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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 $(1-x)^n>1-nx$. ...
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42 views

Find the CDF of the sum of the inverse square of n random normal numbers

Question If I have n independent random normal numbers denoted $X_i$ each with mean $\mu_i$ and variance $\sigma_i$ (for $i = 1 ... n$). For each $X_i$ I have a weighting factor $w_i$. What is the ...
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21 views

How to fit a gaussian model on the overlapped area of two separate gaussians?

Suppose I have two 1D Gaussians distribution with N(U1,S1) and N(U2,S2) where U is the mean and S is the standard deviation. Suppose if we draw these two Gaussians, they overlap on a interval. Now ...
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81 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
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11 views

Finding Conditional Distribution of Multivariate distribution

Q: Supposing y ~ N4 (µ,∑) and µ = (1 2 3 -2)' ∑ =\begin{pmatrix} 4 & 2 & -1 & 2\\ 2&6&3&-2\\ -1&3&5&-4 \\ 2 &-2 ...
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Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
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35 views

Gaussian Bayes Classification with dependent variables..

Gaussian Bayes Classification: two classes: $y \in \{-1,+1\}$ Dependencies for a vector of features ($x_1,x_2,x_3)$: $x_1=z,x_2=2z,x_3=t+3$, where $$P(z\mid y=+1) = \aleph(z;\mu_+,1),\qquad P(z\mid ...
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Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
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50 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
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Probability and continuous distributions

Suppose that the daily consumption of pepsi in ounces is normally distributed with normal(13, 4) in ounces. The daily amount consumed is independent of other days except adjacent days where the ...
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36 views

Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...
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26 views

Sum of two independent truncated gaussians

I'd like to ask for additional info regarding a previous post on the subject: Sum of two truncated gaussian but I can't comment directly on that. Assume $X \sim N(\mu_{1}, \sigma_1^2)$ is doubly ...
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39 views

mean and variance of this Gaussian random variable

I am trying to read through this paper - http://www.malcolmdshuster.com/Pub_2002c_J_scale_scan.pdf Equation 2(b)from the paper says [A] $\nu_k \equiv 2(B_k - b).\epsilon_k - |\epsilon_k|^2 $ where ...
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36 views

Coupling a chi-square to a normal random variable

Let $Z\sim \chi^2(k)$ be a random variable sampled from the Chi-Squared distribution with $k$ degrees of freedom. Vague question: Conditional on the value of $Z$, how can I reconstruct a sequence of ...
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29 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
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20 views

Dice probability normal distribution

You roll a dice 1000 times. Calculate the probability you roll a six between 150 and 200 times. I understand how you calculate this with the binomial distribution: $$ = Binomialcdf(1000, 1/6, 200) - ...
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32 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
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Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
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Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
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T-ratio. Estimation of standard error.

Let $ X = (X_1, ..., X_n)$ be a vector observation collected from Normal Distribution. We don't know neither variance of population nor expected value. We would like to estimate expected value for ...
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Find the distribution of $Z = 1/X_1 + 1/X_2$

Find the distribution of $Z = 1/X_1 + 1/X_2$, where $X_1$ and $X_2$ follow normal distribution. I have $2$ variables with normal distribution, $X_1$ and $X_2$. How can I find the distribution of: ...
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Calculation of arrival time of messages from 1 source through 2 different routes

I need to simulate sending messages from $A$ to $B$ as follows: Each message is sent $N$ times from $A$ on the same time, passes through a certain route $R_n$ and arrives at $B$. Travel time of $R_n$ ...
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34 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
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36 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
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MLE of variance for a spherical Gaussian

I am trying to implement the X-Means clustering algorithm. In it, the authors use the BIC to determine which model fits the data best. It is explained here: ...
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46 views

How good of an approximation is a normal probability distribution for sum of dice rolls?

I want to know how well the normal distribution explains the sum of rolls with n dice with s sides. The mean value and the variance of the dice rolls are $$\mu=n\frac{s+1}{2}$$ and ...
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Does Bivariate Normal have an MLR?

In general, with all parameters unknown I think the answer to this question is no. I think this because in this instance we would have a curved multivariate exponential family. Is this reasoning ...
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48 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
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60 views

Find $E[X+2Y|Z]$

$X,Y$ are independent standard normal. Let $W=X+Y$, $Z=X-Y$. Find $E[X+2Y|Z]$ Attempt: $E[X+2Y|Z=z] = E[X+2Y|X-Y=z] = E[Y+z+2Y] = 3E[Y]+Z = Z$ Is this correct?
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distribution of quadratic form of jointly normal random variables?

I need to derive the distribution of the random variable $\frac{W'(I-1(1'1)^{-1}1')ZZ'(I-1(1'1)^{-1}1')W} {Z'(I-1(1'1)^{-1}1')Z}$ , where $(Z, W)'$ ~ $N(0, I), \,Z=(Z_{1}, ..., Z_{n}), \,W=(W_{1}, ...
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Exponent - Solving for an unknown within an expectation

I have reached a stage where I need to solve for an unknown number, $\theta$ . However, I stuck and don't know how to proceed further. The equation to be solved is: $E\left[ \exp(\theta a^i) * ...
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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 ...
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If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
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Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z ...
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Normal distribution tables - right or left?

Are the probabilities in normal distribution tables given typically to the right or left of the $Z$ score? One such text I am reading says to the right. However, in my lecturer's exercises, I ...
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104 views

Approximating the integral of the exponential of a quadratic function

An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and ...
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Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
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Can a mixture of normals be a constant?

Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?     ...
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If $X|Y$ and $Y$ are both normal, is $X|Y>y$ normal as well?

Consider two random variables, $X$ and $Y$, with the following properties: $X|Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$. Does $X|Y>y$ follow a normal distribution as well? If so, what are its ...
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Question Relating to the Central Limit Theorem

I have the following question: Suppose $X_1,X_2, \ldots, X_{12}$ are identical independent uniform random variables on $[0,1]$. Let the sample mean(lets call it $m$) = $\frac{1}{12}(X_1 + X_2 + ...