# 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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### Probability of constituent distributions [duplicate]

I have a set of N normal distributions representing the points a person can get in a competition. The mean and std deviation of each of these distributions is known, and for simplicity's sake, let's ...
1 vote
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### Prediction intervals of functions of inputs

I have a linear regression model I have estimated which is of the form Y = a + b*X + e. I know how to construct the prediction interval for the outcomes of Y given some value X (say X1), but I was ...
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### Asymptotic distribution in sample variance [closed]

How do you prove that the correct sample variance is asymptotically normal?
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### Statistical test for normality based on quantiles [closed]

Most of the statistical tests of normality that I have found for now are based on samples (instead of quantiles). Having implemented a graphical comparison approach, I am currently in search of a ...
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### Mathematical Theory of Monotone Transforms

For $n$ observations $X_1, \ldots, X_n$ of one sample following distribution $P$ chosen from population $\mathcal{P}$, suppose that we observe $X_i = x_i \in \mathbb{R}$. Empirically, we may detect ...
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### Lower bound of KL divergence of Gaussian mixture with Gaussian (univariate)

I'm interested in a non-zero Kullback-Leibler divergence lower bound between a Gaussian mixture $q$ and a standard Gaussian $p(x) = \mathcal N(x \mid 0,1)$, both univariate: $D_{KL}(q \parallel p)$. I'...
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### Distribution after adjusting sample standard deviation [closed]

Suppose you repeat measurements of continuous values 3 times for every observation, and you have 10000 different observations. For every observation, you determine the mean and the sample standard ...
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### Maximum Likelihood Estimation with Gradient Descent and Squarred Loss

My goal is to learn parameters $\mu$ and $\sigma$ of a univariate Gaussian distribution using gradient descent to validate my understanding of the algorithm by deriving all the formulas from scratch. ...
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### Finding the mean and variance of a random variable that is an affine function of a Gaussian random variable [closed]

The question: The catch here is I can't just calculate expectation and variance of Y and insert it in the Gaussian formula. I need to use change of variable and derive this relation.
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### Is it better to use the W or p-value to determine normality using Shapiro-Wilk?

I have read somewhere that W values above 0.9 are considered normal. So I wanted to use that as a cutoff for normality. However, upon running various scenarios I have come up with the result of W = 0....
1 vote
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### Ratio of Normal pdf to cdf

I want to show that $$\Bigl\lvert \frac{\phi(a)}{\Phi(a)} - \frac{\phi(b)}{\Phi(b)} \Bigr\rvert \leq |a-b|$$ where $\phi$ is the standard normal pdf, and $\Phi$ is the standard normal cdf.
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### Distribution after Combining Two Sets of Normal Distribution Samples

Suppose I draw $N1$ samples from distribution $N(\mu_1,\sigma_1^2)$, $N2$ samples from distribution $N(\mu_2,\sigma_2^2)$. These two distributions are independent. Can the combined sample of $N1+N2$ ...
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### How to generate biased random walks from data?

Imagine I have data (as a list of lists) of the following kind, where I am tracking the positions of specific object centroids over time (independent form each other, so the initial location does not ...
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### What is the distribution of a ratio of two normal distributions? [duplicate]

Assume that I have a resistor with an unknown resistance R. I measure the current trough it, I, and the voltage across it, U. I can then estimate R: $$\hat{R} = \frac{U}{I}$$ Let us assume I get the ...
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### Characterization of (Stationary) Gaussian Processes [closed]

There are many posts about characterizations of Gaussian distributions, such as here and here. These provide interesting results of the form: "If a random variable/distribution satisfies X,Y,....,...
1 vote
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### Interpretation of generalized eigenvalues summary, matrix distance, information geometry

This question is about how generalized eigenvalues of two covariance matrices relate to the discriminability of their associated Gaussian distributions. The generalized eigenvalue problem for two ...
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### When conditioning a Gaussian process, why is the conditional mean always trending upwards?

I have a Gaussian vector with mean $\mu$ and covariance matrix $\Sigma$, both estimated. The vector $\mu$ represents a process and for each entry starting with the second, I want to find the ...
1 vote
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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 ...
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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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### 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 ...
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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~...
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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 $G_i$...
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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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### 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 ...
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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" ...
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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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### 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 ...
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### 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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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\ ...