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

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0answers
6 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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0answers
7 views

Is a circular and spherical gaussian the same thing?

This pdf https://www.cs.princeton.edu/courses/archive/spring07/cos424/scribe_notes/0419.pdf mentions "circular gaussians" as the simplest gaussian in 2 dimensions, what you get if you have I as the ...
2
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2answers
60 views

Central Limit Theorem, why $n \ge 30$?

This is what I think the technical statement of CLT is: If we consider $\overline{X}_{n}$ coming from a sample of $\mathcal n$ independent and identically distributed random variables $X_{i}$ with ...
1
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1answer
13 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
1
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1answer
24 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
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0answers
9 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
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0answers
10 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
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0answers
11 views

Approximate a function with a gaussian distribution.

I have a function which has a bell-type graph and i need to find a Gaussian(Normal) with the appropriate mean, variance and constant factor which is close to the original function.The function in ...
0
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0answers
19 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
0
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0answers
31 views

Normal distribution where variance depends on mean

Let $x = \bar{x} + \epsilon$ where $\bar{x} \sim \mathcal{N}(\mu,\sigma^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2(\bar{x}))$ are independent, i.e., the expected value of $x$ is normally ...
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3answers
35 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
0
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1answer
28 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
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1answer
45 views

Does a conditional normal distribution imply an unconditional normal distribution?

I have often seen it claimed that for scalar random variables $y$ and $x$, the conditional normal distribution $$ y\mid x\sim N(0,x^2) $$ also implies the unconditional normal distribution $$ ...
1
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0answers
5 views

Conditional Covariance of a Normal conditionally autoregressive (CAR) prior

Suppose $$\bf{\nu}|\mu,\rho,\delta^2 \sim N_p(\mu \bf{1},\delta^2(D_w-\rho W)^{-1}),$$ where $W$ is a binary symmetric matrix and $D_w$ is diagonal with $(D_w)_{ii}=\sum_j w_{ij}$, $\mu$ is a scalar. ...
3
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1answer
35 views

Distribution of sum of product-normal distributions.

I know that the distribution of a product $Z=XY$ of two normally distributed variates $X$ and $Y$ with zero means is the product normal distribution [Mathworld]. What is the distribution of $Q=\sum ...
0
votes
1answer
21 views

Is this function increasing? (standard normal distribution)

Where $\phi\left(z\right)$ and $\Phi\left(z\right)$ represent the standard normal pdf and cdf respectively. 1) Is the function $f\left(z\right)=\dfrac{\phi\left(z\right)}{1-\Phi\left(z\right)}$ ...
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2answers
38 views

Help me prove this an absoulte inequality [closed]

Let $0\leq\pi_{i}\leq1\quad s.t. \sum\pi_{i}=1$, and $f(x;\phi_{i})$ be a normal pdf paired with $\pi_{i}$ Then, I need to prove below $\sum\pi_{i}f(x;\phi_{i})^2\geq(\sum\pi_{i}f(x;\phi_{i}))^2$ ...
0
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0answers
15 views

Normal Distribution Analysis

I have listed the problem below! I solved this problem using quartiles but for some reason I am not getting the correct answer. For part A, I used a guess and check method using quartiles and got ...
0
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1answer
25 views

Integrate complicated function

Can someone help me write this in matlab? (I just need to transfer the formula to matlab format so I integrate it): $$\int_{point(i)}^{point(i+1)}((x-c(i))^2*normpdf(x, -0.04, sqrt(0.11)))$$ ...
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0answers
20 views

Updating distribution with new samples

If I have a specific distribution as a belief state and real distribution, which is unknown. After that I sample $n$ times from real distribution(which is unknown). How I can update my belief ...
0
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1answer
109 views

Combined Likelihood for probability [closed]

The measurement errors made by a measuring device operating under two different conditions, $i = 1$ and $i = 2$, are normally distributed with mean $0$ and variance $\sigma_i^2$. You may assume that ...
7
votes
1answer
109 views

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
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0answers
19 views

Rearrange for solving x (Miller & Siegmund, 1982; equation 8)

I have the following formula from a paper back in 1982 by Miller & Siegmund, "Maximally Selected Chi Square Statistics": α = 0.05 φ() is the standard normal density function: Everything else ...
1
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1answer
24 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
0
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1answer
15 views

How can you normalize two data sets to the same scale?

I have two data sets, one that ranges from 0-200, and another that ranges from ~400-~2500. I would like to compare the two according to a score from 0-10. I remember about normalizing from a ...
0
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0answers
15 views

Non-Geometric Proof of Random Normal Projection Identity

Many papers on locality sensitive hashing, sketching and similar use the following lemma: If $r\in\mathbb{R}^d$ is a random vector with all entries normally, independently distributed as ...
0
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0answers
12 views

central limit theorem and sampling dist.

If you takes samples from a distribution, and you can see that they have different variances, can the central limit theorem still be applied. The computer vision teqnique i am referring to is this ...
-1
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0answers
9 views

Normalizing histograms 3 D

We downsample each of Y,Cb and Cr channels into 64 levels.Then we can construct a three-dimensional color histogram in which each dimension has 64 bins. We normalize the color histogram in ...
0
votes
2answers
30 views

Robust estimator

What does it mean that an estimator is robust? How can you tell whether an estimator is robust or not in statistics? I need to discuss whether the maximum likelihood estimators of the normal ...
0
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0answers
9 views

Different continuous distributions applicable to crowd scenarios

I working on different distributions of crowd exiting from a subway train. Right now I have studied normal distribution, that is centred around the exit. What could some other such distributions that ...
1
vote
1answer
51 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
2
votes
1answer
31 views

Intuition for proof of Slepians Inequality

If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E ...
0
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0answers
23 views

Normal distribution function:determine probability of a given point in Java

My statistics since high school is gone An I am struggling to find out a way to determine the probability of a given point in a Normal distribution in java. I see that Colt ...
1
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0answers
31 views

Show Almost Certain Convergence of a Sequence of Normal Random Variables

Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$ I am aware of the ...
0
votes
1answer
36 views

Integral of normal distribution curve

I am having hoping to use the integral of the normal distribution curve to find the probability of having a mean of $0.30$ or greater, i.e. one tailed distribution. With a sample standard deviation of ...
0
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0answers
10 views

Chi sqaured table for degrees of freedom 616?

In order to check heteroskedasticity, we use the White's test. I tried to follow this method below, however, could not find a table with df=2016 and 95,5% confidence. I don't understand how we get ...
0
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0answers
14 views

Product of two gaussion functions [duplicate]

I would like to ask how to calculate the product of two Gaussion functions? As far as I have understand the product is another Gaussion function. How can this be? I wonder if the std.dev. will be ...
0
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0answers
25 views

Determination of a Lower Confidence Bound

Two stochastic variables $X$ & $Y$are normally distributed with $X\sim \mathcal{N} (0,1)$ and $Y\sim \mathcal{N} (2,1)$. We declare that $W = X + Y$ and $W$ is normally distributed with $W\sim ...
1
vote
1answer
26 views

Standard Normal Distribution Transformation Z=lnY

I'm not sure if my approach to this problem is correct and I need help I need to apply $Z=\ln{Y}$ to the following standard normal distribution and then find the distribution of $Y$ ...
1
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1answer
29 views
2
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1answer
16 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
2
votes
2answers
36 views

Probability that $x > 0$ given that $x > y$ for independent and normally distributed $x,y$

I was recently been asked this question in an interview but not able to solve it as I am rusted in Bayesian conditional probability. Here is the question: $x$ and $y$ are independent variables ...
2
votes
1answer
34 views

Why is the $0$th percentile of the standard normal distribution $-\infty$?

Why is the $0$th percentile of the standard normal distribution $-\infty$? I can't explain the cause except saying there is no area under the curve. So it goes beyond the bell-shaped curve.
-1
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1answer
40 views

Independence of Gaussian Distribution Function with Different Means

Is there any way to prove that $N$ Gaussian distribution functions are linearly independent if and only if the means are different. For example, if ...
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0answers
19 views

Standard Deviation of Binomial vs Sampling Distribution

High school dropouts make up 14.1% of all Americans aged 18 to 24. A vocational school that wants to attract dropouts mails an advertising flyer to 25,000 persons between the ages of 18 and 24. (a) If ...
1
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1answer
41 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
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votes
0answers
43 views

For a given range of prices which follow a normal curve, what is the probability of a certain price?

If prices of a commodity in the market a following normal distribution but will stay within the range of $m$ (minimum price) and $M$ (maximum price). What is the probability that a certain price $a$ ...
0
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2answers
21 views

How to analytically find probability of certain combinations between normally distributed populations?

Let's say we have two normally distributed populations A and B, which have different means and standard deviations. We then pick one item from population A and one from population B. How could we ...
2
votes
1answer
29 views

Is the following process bounded (iterative normal sampling)

We define the following stochastic process: $X_0=1$ $\forall i\geq i:X_i\sim\mathcal N(0,X_{i-1}^2)$ That is, we first sample $X_1$ from the normal distribution with variance $1$, then in the ...
0
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0answers
18 views

Effect of Normalization On a Distribution

Let's assume that we have calculated some values proportional to probabilities for a special distribution(some hypothetical distribution other than normal distribution but we don't know which ...