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Questions tagged [normal-distribution]

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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How to derive Diffusion Model's reverse conditional probability when it's tractable via conditioning on $x_0$

Can anyone help me with understanding how the $\tilde{\beta}$ and ${\tilde\mu_t{(x_t, x_0)}}$ are derived? It seems to me that exponential term is a 2nd order polynomial term and it doesn't really ...
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log transform left data in r

I am having trouble finding the transformation operation for left/negatively skewed data. The catch? All of my values are between 0 and 1. As such, trying the standard log10 transformation command ...
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Equivalence of inverse transformations under distributional equivalence

Consider continuous, invertible transformations $g,h : \Bbb{R}^d \rightarrow \Bbb{R}^d$ and suppose $g(Y) \overset{d}{=} h(Y)$, where $Y$ is a $N(0, I)$ random variable. Then what can we infer about ...
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Is it possible for some p-values to be impossible? (because statistic generated by parametric bootstrap is mostly the same value.)

I am using a parametric bootstrap/monte carlo hypothesis testing method to generate the null distribution of the log likelihood ratio statistic. However, I am worried I might be doing it wrong ...
A Friendly Fish's user avatar
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Wasserstein distance to assess the degree of normality

The Wasserstein distance between two probability measures with quantile functions $F^{-1}$ and $G^{-1}$ is given by \begin{align} W(F,G) = \int_{[0,1]} |F^{-1}(t) - G^{-1}(t)|dt \end{align} Now let's ...
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Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n

For random variables $X_1, \cdots, X_n$, we denote the order statistics by \begin{align} X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt] X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\ &...
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Reparametrization trick in GMM

Assume I have two random variables, $X$ and $Y$. If $p(Y|X)=\mathcal{N}(Y;\alpha.X,\sigma^2\mathbb{I})$, I can calculate $Y$ by using reparametrization trick: $Y=\alpha.X+\sigma.\epsilon$, with $\...
Toan Le's user avatar
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How do companies decide the warranty period for a product?

$\text{Question:}$ A firm produces machines with a lifespan, whose distribution has a mean of $\mu$ months and standard deviation of $\sigma$ months. The firm wishes to introduce a warranty scheme in ...
Samar's user avatar
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Worst-case analysis using confidence intervals

I need to state something about the minimum expected value of the population with a certain confidence, and I'm not sure if I understand the process correctly. Say I have a process with a strict ...
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Posterior expectation of normal distribution with "truncated" observation

Consider the following problem of estimating an unknown parameter from normal samples: Suppose that $\theta \sim N(0, \tau_\theta^{-1})$, where $\tau_\theta \ge 0$ is the prior precision. Consider two ...
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Testing for Gaussian Mixtures

Suppose I observe data from a univariate distribution $\mathcal{D}$ with zero mean, and unit variance. We know that either $\mathcal{D} = \mathcal{D}_0 = \mathcal{N}(0,1)$ is standard Gaussian or $\...
Claudio Moneo's user avatar
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Intuition behind relation between Gamma and Standard Normal distribution

I read if $Z$ is a random variable with a standard Normal distribution and $X=Z^2$ then $X \sim \operatorname{Gamma}(1/2, 1/2)$. I understand the math (manipulations of formulas) behind it. What about ...
Gabriele Bettineschi's user avatar
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Distribution of the model vs. Distribution of the Residuals

Let's say I'm going to do an analysis where my response variable has a gamma distribution. I perform the analysis pointing to the distribution in my model (eg. using the lme4 package, m1<-glmer(Y~...
Graciliano Santos's user avatar
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Bound Product of Independent Gaussians

I'm interested in obtaining upper bounds on $$ \Pr[\prod_{i\in[n]}|G_i| > x] $$ where $G_i\sim\mathcal{N}(0,1)$ i.i.d, and $[n] := \{0,1,\dots,n-1\}$. The most naive bound is to note that each $G_i$...
Mark Schultz-Wu's user avatar
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What is the meaning of term "whiten" data in relation to Mahalanobis Distance?

I'm writing my thesis and I have trouble in understanding the paper: https://people.bu.edu/bkulis/pubs/ftml_metric_learning.pdf My major is not mathematics, but I can understand the basic so I hope ...
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Testing difference between regression coefficients for different greoups

I am aware that this questions was addressed several times in this site and that there are certain papers discussing this issue for instance 1, 2. USING THE CORRECT STATISTICAL TEST FOR THE EQUALITY ...
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Rough answer for the maximum of absolute value of $n$ standard gaussians (Computer Age Statistical Inference Problem 1.3)

I am working through "Computer Age Statistical Inference" as a self-study and am stuck on the follow exercise (1.3): The details of equation 1.6 are unimportant for the exercise, so far as ...
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2 answers
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How can I reconstruct a normal distribution from a set of percentiles?

I have the 3rd, 10th, 50th, 90th, and 97th percentile values of a normally distributed variable, and I wish to generate a dataset that will allow me to query for other percentile values (say, the 67th....
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Conditional expectation of a multivariate normal distribution

Let $(X,Y,Z)$ have a multivariate normal distribution: \begin{align} (X, Y, Z) \sim N\left(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & \rho_{xy} & \rho_{xz} \\ \rho_{xy} &...
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Non-Parametric Two Way Within-Subjects ANOVA

I have a dataset with 4 individuals that are measured twice in each of the 5 groups (so in total 40 observations). Subject ID Group Value 1 1 A 45 1 2 A 62 1 1 B 70 1 2 B 37 ... ... ... ... 4 2 ...
mschal's user avatar
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4 votes
2 answers
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How to compare three groups of patients if one group is normally distributed but the other two are not?

Retrospective study involving three groups of patients. Group 1 are patient's diagnosed with sepsis. Group 2 are patient's diagnosed with a localized infection. Group 3 are "healthy" ...
Faye Morton's user avatar
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Precision of estimates of lower bit error probabilities at higher SNR

For my university lab in wireless communications, I simulated a simple uncoded BPSK (binary phase shift keying) channel with AWGN (additive white gaussian noise) to estimate the BER (bit error rate) ...
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Questions on Basic Data Cleansing For Linear Regression

I'm following some tutorials on doing some linear regression and as I was building my notebook, I'm working on outlier detection and amongst the techniques described for doing outlier detection, one ...
joesan's user avatar
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Product of RVs of Which Distribution Approximates Normal Well?

Suppose that I have $N$ i.i.d. random variables with a distribution $Q$, which has mean around 1. That is $R_1, R_2,\ldots,R_N \sim Q$. I would like $\prod_{i=1}^N R_i$ to approximate a normal random ...
Cagdas Ozgenc's user avatar
1 vote
1 answer
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Distribution of a spread of observations in triplicate sample taken from Gaussian distribution

Suppose random triplicate samples are taken from a Gaussian distribution with known mean and SD. What should be the distribution of the maximum absolute difference between 3 possible pairs of ...
Maciej Tomczak's user avatar
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2 answers
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When using Normal approximations of human height, what's the probability two people of unknown but approximately equal height are both tall?

Basically, I saw a photo of two people with unknown approximately equal heights and was struck with the thought "How likely is it that they're both tall?". That's the problem I want to ...
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6 votes
2 answers
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How to write a function for the normal copula in R?

How can I write the following function for the normal copula in R? $$ C_\theta(u, v)=\Phi_\theta\left(\Phi^{-1}(u), \Phi^{-1}(v)\right), $$ where $\Phi$ is the $N(0,1)$ cdf, $\Phi^{-1}$ is the ...
Aria's user avatar
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Exponential family expression for bivariate Gaussian

Consider a bivariate Gaussian distribution with the following density: $$f(x,y) = \frac{1}{2\pi \sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2)}$$ How can we write it in the exponential ...
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5 votes
2 answers
368 views

Correction for heavy-tailed distribution of residuals?

I'm interested in studying the effect of $x$ on $y$ using a fixed effects method. The residuals follow a heavy tail distribution, as the normal Q-Q plot suggests. For inference, I need a normal ...
TFT's user avatar
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5 votes
3 answers
625 views

For what kind of distributions could the joint distribution be determined uniquely by marginal distribution and correlation?

Assume $X$ and $Y$ are from the same distribution $P$ and $\rho = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$ is fixed. For what kind of $P$ can we determine uniquely the joint distribution of $X,Y$? I know ...
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Why confidence interval for proportion uses z instead of t score? [duplicate]

The confidence interval (Wald interval) for the parameter $p$ of a binomial distribution is computed from the approximation by a Normal distribution: $$ \ p ~~ \approx ~~ \hat p \pm \frac{\; z_\alpha\ ...
luchonacho's user avatar
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3 votes
1 answer
108 views

Uniform distribution over a triangle

Problem Consider a triangle $T$ with vertices $V_1,V_2,V_3 \in \mathbb{R}^2$ and let \begin{equation*}\begin{aligned} y&=z+v\\ v&\sim\mathcal{N}(0, R)\\ z&\sim\mathcal{U}(T) \end{aligned}\...
matteogost's user avatar
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Strange Variance Term for Normal Prior $w^2\sigma^2$

I've attached two screenshots, one with the question and one with the answer. It seems to me that the prior is wrong and it should include $w^2$ not $w^2\sigma^2$ I apologise for, including such a ...
CormJack's user avatar
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0 answers
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Calculating the mean and standard deviation from a normal distribution [duplicate]

I am interested in methods for finding the mean and standard deviation of a normal distribution. It is inspired by the following question that is part of the A-level maths course in England. The ...
Bysshed's user avatar
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0 answers
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Uniform density over 2 segments [duplicate]

Background Let $V_1, V_2 \in \mathbb{R}^2$ be the vertices of a segment and let $z$ be uniformly distributed over that segment. Now consider the random vector \begin{equation*} \begin{aligned} y&=...
matteogost's user avatar
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1 answer
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How to choose between gamma and Gaussian given a choice of gauges?

I'm trying to make the choice between the gamma and Gaussian distributions as a prior distribution for some data. When I learned statistics a while ago, I was given the rule of thumb: if your data ...
Corbin's user avatar
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3 votes
0 answers
61 views

Probability that normally distributed variables will have a specific ranking

There are $k$ players playing a game, each gives a performance $X_k \sim N(\mu_k, 1)$ and we observe their ranking from best to worst (a permutation of the player indexes). How to calculate the ...
fhucho's user avatar
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0 votes
1 answer
21 views

Reverse distribution in DDPMs

In Denoising Diffusion Probabilistic Models, we want to reverse a forward process where we blend gaussian noise into a data point $\mathbf{x}_0$ over $T$ steps. The result of each forward step applied ...
CoconutFred's user avatar
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Is it possible for two normally distributed (ideally nondegenerate) random variables with 0 covariance to not be independent? [duplicate]

I know if the variables are jointly normal, 0 covariance implies independence, and I know that marginal normal distributions don't imply a joint normal distribution, but is there a way for the ...
Yash Permalla's user avatar
1 vote
0 answers
75 views

My professor insists that this data is normally distributed but I cannot work out how that can be!

I have a data set on deforestation and the impact that the distance from a road has on deforestation. For the assignment, I need to assess the correlation between these two variables. In class, my ...
Daisy's user avatar
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5 votes
2 answers
411 views

ANOVA vs Kruskal Wallis - Small sample size

I have data from an experiment comparing plant weights for 4 independent treatment groups. The data seem to be normally distributed (I have been warned about using statistical tests for normality). ...
user411569's user avatar
1 vote
0 answers
57 views

Welch ANOVA for Comparisons of Elevation & altered Sediment Accretion of Sea Ecosystems: Large DEM with no normal distribution & heterogeneity

I am investigating the elevation characteristics and sediment accretion effects of two distinct ecosystems within the Wadden Sea, impacted by the bioinvasion of one species (ecosystem one) and ...
OmteK.'s user avatar
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5 votes
2 answers
495 views

Detect rare high-value measurements in a series of measurements

We do a measurement on 1000 samples to detect if a chemical element A is present, and for each measurement, two cases can happen : the element A is not present, and the values we get are a "...
Basj's user avatar
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1 vote
2 answers
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Distribution of transformed normal [closed]

Suppose $X$ is normal with mean $\mu$ and standard deviation $\sigma$ and $\Phi$ is the standard normal distribution function. What is the distribution of $\Phi(X)$?
John L's user avatar
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Confidence interval for a normal RV that its std is the mean of a different normal RV

Question: Suppose $X\sim\mathcal{N}(\mu_X,a),Y\sim\mathcal{N}(\mu_Y,b\cdot\mu_X)$ where $a$ and $b$ are known, but $\mu_X$ and $\mu_Y$ are not. What confidence bounds can we give for $\mu_Y$ from one ...
Amir's user avatar
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5 votes
2 answers
135 views

Why do many people like to use "normal distribution" instead of $\beta$ distribution when the theoretical range of data is finite?

Say we are approximating the distribution of the test score of a large group of students, while the theoretical minimum score is 0 and the maximum is 100. In this case, many people use the normal ...
Zuriel's user avatar
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1 vote
1 answer
54 views

What is the distribution of the unbiased estimator of variance for normally distributed variables?

I must be making some mistake in my derivation of the distribution of the unbiased variance estimator for i.i.d. $X_{i} \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$. We have $\bar{X} =\frac{1}{n}\sum\...
YEp d's user avatar
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3 votes
1 answer
41 views

Is every method of moments estimator (MME) asymptotically normal distributed?

As already written in the question, I am asking myself whether every method of moments estimator (MME) is asymptotically normal distributed? Formulated differently, is every (central or non-central) ...
user avatar
1 vote
1 answer
84 views

How to calculate and plot a confidence interval for a bivariate normal distribution?

I have a two-dimensional normal distribution (with correlations). I do not have data points, but only the 2D mean and the covariance matrix. I want to draw the 68, 95 and 99% confidence intervals ...
tpd's user avatar
  • 111
0 votes
1 answer
23 views

How to Find Probability Sum of Squares of Std Normal is greater than Sum of Squares of Non-Standard Normal with Mean 0

I'm looking for the probability that the sum of squares of a standard normal is less than the sum of squares of a non-standard normal with mean 0 and fixed std-dev. Lets say $X_i \sim \mathcal{N}(0,1)$...
CatGod's user avatar
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