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The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is one of the most important distributions in statistics. Use the [normality] tag for asking about testing for normality.

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Normal Distribution Help [on hold]

Here is the question "The gross weight of a large bag of landscaping rocks averages 50 pounds with a variance of 25 pounds2. What is the chance that 10 randomly chosen bags weigh a total of at least ...
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1answer
34 views

How to get the distribution of $\frac{1}{n} \sum_i x_i^2$

$X$ follows a normal distribution $X \text{~} N(\mu, \sigma^2) $. And there are $n$ samples. Then what is the distribution of $$\frac{1}{n} \sum_i x_i^2$$ I do understand $\frac{\sum_i x_i}{n} \...
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What is the mean? Would 5:45 be the center value? [on hold]

Car accidents have a normal distribution between 4:30 and 7 pm. what is the probability of having car accident after 6. I tried assuming the mean was 6, so the value would be zero on the z table, ...
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1answer
19 views

What is the correct distribution of reading time for short text

A mobile app I am creating shows a sequence of headlines which when tapped on shows more detailed information. The detailed information can belong to one of several, but small, categories and has a ...
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7 views

Relating Means/Stds Between Gaussian and Rayleigh Distributions

suppose I have something like a targeting problem, where I specify an angular dispersion in the up and down direction with two gaussian distributions, each having a mean of 0 and a std of 0.3 degrees. ...
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31 views

Difference between Gaussian process regression and other regression techniques (say linear regression)

I am confused about the differences in the regression techniques available. Take for example, linear regression. In this case, we construct a model $y = \beta^Tx + \epsilon$ where $\epsilon \sim N(0,\...
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38 views

Confidence regions on bivariate normal distributions using $\hat{\Sigma}_{MLE}$ or $\mathbf{S}$

Given a $5 \times 2$ dataset $\mathbf{X} =\left( \begin{array}{rr}-0.9&0.2\\2.4&0.7\\-1.4&1.0\\2.9&-0.5\\2.0&-1.0 \end{array} \right)$. Assume that $X\sim N_2(\mu, \Sigma)$. ...
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17 views

How to calculate Expectation of Multivariate Gaussian with x drawn from another Gaussian distribution?

Say, there is a $n$-dimension multivariate Gaussian, $g(x) = N(x:\mu, \Sigma)$ where $\mu$ is $n$-dim mean vector, and $\Sigma$ is $n \times n$-dim covariance matrix. I would like to calculate "...
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1answer
32 views

Normal Distribution using Z - Score Rules

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 ...
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13 views

Combining GMM's in different dimensions

I have input $X$, which follows a distribution $P(X)$, which is best modeled using a mixture of Gaussians. I also have another random variable $T$, which is also best modeled by a mixture of Gaussians....
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1answer
14 views

Distributional assumption for a VAR model: is normality needed?

Do all variables in a VAR (Vector Autoregressive model) need to be normally distributed? Or there is no restriction about the distributions of the variables in this model (normal or otherwise)?
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Marginal distribution of normal random variable with a normal mean

I have a question about calculation of conditional density of two normal distributions. I have random variables $X|M \sim \text{N}(M,\sigma^2)$ and $M \sim \text{N}(\theta, s^2)$, with conditional ...
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17 views

MLE of $\mu_k$ and $\Sigma$ for the singular matrix $\{\mu_k \}_1^k$

This problem is number 4.8 from the Elements of Statistical Learning by Hastie, Tibshirani, and Friedman. Consider the multivariate Gaussian model $(X|G = k)\sim N(\mu_k,\Sigma)$, where $k$ is one ...
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23 views

Residuals Diagnostics Forecasting principales [on hold]

Help me please Are the following statements true or false? Explain your answer. Good forecast methods should have normally distributed residuals. A model with small residuals will give good ...
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42 views
+50

Deriving predictive distribution

In Bayesian Regression, I am confused how to to get $f*$ and $\sigma*$, given $$y^∗ \mid \vec{y}\sim\mathcal{N}(f^∗ , σ^∗ )$$ $$ p(y^* \mid \vec{y}) = \int{p(y^* \mid \vec{w}) p(\vec{w} \mid \vec{y})...
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41 views

Ratio of mean absolute deviation to standard deviation under normal distribution

Can someone show why the ratio is $\sqrt{\frac{2}{\pi}}$ ?
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1answer
47 views

Statistical analysis on confidence intervals

I have a data set where the data, when plotted, is not normal. Log-transforming the data makes it normal. Should confidence intervals for the population mean and hypotheses testing about the ...
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1answer
42 views

How to calculate the integral of Normal CDF and Normal PDF?

I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where $A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
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47 views

Sampling without replacement - Normal sampling distribution [duplicate]

Most of the introductory stats textbooks, treat the sampling distribution of the mean as a normal distribution when sampling is done without replacement and n/N > 0.1. They just use of the finite ...
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0answers
13 views

How to estimate contribution of noise sources

We're trying to estimate the contribution of a device on a performance indicator on the quality of transmission of some signal. The value for performance indicator is assumed to be normally ...
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0answers
21 views

Variance of MLE for the common mean of two normals

I have the following problem. $X_i \overset{IID}{\sim} Normal(\mu, \sigma_1^2) $, $Y_j \overset{IID}{\sim} Normal(\mu, \sigma_2^2), i = 1, \cdots, m, j=1, \cdots n $ Find the MLE for the $\hat{...
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How to find value of $\vert$$z$$\vert$ in normal distribution

Given that for a standard normal variable $Z$,$p(0<z<0.8) =0.2881$ The value of $p($$\vert$$z$$\vert$ $\geq$$0.8)=?$ I already know how to find $p(z$$\geq$$0.8)$ which is equal to $0.21186$. ...
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Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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Gaussian distribution fit lab data, via matlab. How to find varaince? [closed]

I am currently doing an experiment in gamma ray spectroscopy and trying to find the standard deviation of the Gaussian fit I have made to the data. The issue is matlab provides me with the confidence ...
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How do I add a standard deviation scaling factor to my cumulative normal distribution function in R? [migrated]

I am using the quickpsy package in R. I would like to have a parameter that scales the standard deviation that quickpsy calculates for the cumulative normal distribution function ...
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1answer
44 views

Finding MLE of the common $\mu$ from normal samples with two unknown variances

My problem is as follows, Find the maximum likelihood estimator, $\mu^{MLE}$ from $(m+n)$ samples, where $X_1, \cdots, X_m \sim N(\mu, \sigma_1^2), Y_1, \cdots, Y_n \sim N(\mu, \sigma_2^2),$ where ...
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39 views

What do machine learning people mean by $x \sim X$? [closed]

See the question I posted on math stackexchange. https://math.stackexchange.com/questions/2947908/what-does-x-sim-x-mean-in-probability/2947934?noredirect=1#comment6087816_2947934 I just want to ...
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17 views

How to integrate probability density of sum of two indepedent random variables with a finite lower bound on one of them?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$ but $$u = \ln(A)-C-k$$ where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant. I am ...
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1answer
29 views

How can a dnorm probability be larger than a corresponding cumulated pnorm probability [duplicate]

Using the probability distribution density functions dbinom, dnorm, etc. and the corresponding cumulative probability functions pbinom and pnorm, I noticed that the dnorm density values could be ...
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3answers
102 views

what does p( y | μ,σ²) really mean?

Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically: I understand what p( A | B ) where A="I am sick" and ...
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1answer
27 views

Any transformation of Gaussian random variables admits covariance calculation after transformation?

Sigmoid transform of Gaussian random variables does not have an easy calculation of covariance. Specifically, I'm looking for a function transform $$f: \mathbb{R}\rightarrow [0, 1]$$ such that it ...
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1answer
19 views

Multiplying two event with probability density function, is it possible?

in my exercise, $X$ is the size of a tree trunk, and $X$ follows a normal distribution $\mathcal{N}(9,0.4)$, we want to know $P(8.8\le X\le11.2)$ So I though that I could do this: $P(8.8\le X \le11.2)...
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1answer
61 views

Compute the mean of normalized norms of linear transformations of Gaussian random vectors

if $M$ is a $m\times n$ constant matrix and $\eta\sim\mathcal{N}(0,I)$, then does $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M\eta\rVert}{\lVert\eta\rVert}\right]$$ exist? Also, let $x\in \...
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1answer
28 views

multivariate normal distribution range [duplicate]

Simple question about MVN pdf. I understand the domain to be [0,1]. However, why does scipy.stats.multivariate_normal.pdf output values above this range. E.g. <...
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1answer
38 views

Help with notation p(y|x, w, beta)

In a Machine Learning course the notation used is the following: I understand the normal (I think) on the RHS, but I can't figure out what this p is. Is it: ...
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Would randomly permuting a quasirandom sequence successfully avoid otherwise intrinsic correlations in estimation?

Let $A$ be an $n \times n$ (Ginibre) matrix of complex-valued entries, the real and imaginary parts of which are, thus, standard normal variates, and $U$ be an independent such matrix, the rows and ...
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Var(M hat) for inverse-variance weighting with over-confident estimates

I'm using inverse-variance weighting to combine two estimates of a given data point, from different sources. Calculating the M hat is going just fine. However, what I'm running into is that the ...
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10 views

How tailed area decided for hypothesis testing?

I have a doubt in basic hypothesis testing. Suppose I have a testing setup as below. I have assumed null hypothesis that my sampling distribution has $\mu = 60$, and I get a sample set with sample ...
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1answer
68 views

Is there a method to transform all z-scores into positive values?

During my calculations I need to use square roots but z-scores can be negative. Is there a trick to transform them into positive value without missing the usefulness of z-scores? What if I have not ...
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20 views

What is the logic behind the calculation of sample size in the attached problem?

how to arrive at this n=543..till 276.87 logic looks good. Reference Problem 7-62 Statistics for management Levin,Rubin,Rastogi,Siddiqui.
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47 views

What is this parameter estimation strategy called?

Let $X_1, X_2, \ldots, X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Consider the problem of estimating $P(X > 100)$. One way to accomplish this is ...
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$V=XY/\sqrt{(X^2+Y^2)}$ Distribution [duplicate]

If $X$ and $Y$ are iid N(0,1) what is the pdf of $V=XY/\sqrt{(X^2+Y^2)}$. I have found out the distribution of $1/(X^2)$. So will that be any useful and if yes how ?
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1answer
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How to “normalize” standard deviations?

I'm a computer science guy who's recently moved into Performance Engineering. As part of this job, I now find myself needing to analyze results of tests (duh). However, my lack of statistical ...
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Distribution of $X_1X_2+X_3X_4$ [duplicate]

Let $X_1,X_2,X_3,X_4$ be independent $N(0,1)$. Find the distribution of $Y= X_1X_2+X_3X_4$ I am having difficulty in solving this question or atleast how to approach it.
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Solution of $D_{KL}(q_\phi(z)||p_\theta(z))$ of Gaussian case

Following is from the original paper of concept of VAE(variational autoencoder) by Kingma,Welling 2014 B. Solution of $D_{KL}(q_\phi(z)||p_\theta(z))$ of Gaussian case The variational lower bound (...
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1answer
29 views

95% chance of 1.6 Standard deviation?

I'M reading a book on portfolio management and I can't understand how the author came up with 95% chance of volatility reaching 1.6 times..here's the excerpt: "Suppose that we were to forecast that ...
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16 views

Normal to binomial solved clarification [duplicate]

The mass production of pens result in there being one defective pen in 20 pens on average. Find the probability that in batch of 300 pens there will be 24 or more defective? What I did: Used ...
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1answer
43 views

Find the peak of probability 30

If probability is P=1/3 and N=100, what is the probability you succeed 30 times? where does this P30 fall in the curve? I don't have mean or standard deviation to calculate the height of the point!
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51 views

Standard normal intuitive understanding

What does it mean for a standard normal to have mean 0 and standard deviation 1? I'm having trouble understanding - what is a "normal variable"?
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How do I find the mean of Z = Discrete RV + a Gaussian RV?

I am asked to find mean and variance of Z. (image) Since I am solving a preparatory examen to study, it is not clear to me how to approach the topic because I don't understand the question correctly. ...