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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Normal density's rate of convergence to 0 as mean goes to infinity while x and standard deviation are fixed

Consider the density of the Normal distribution given by $$f(x; \mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x - \mu}{\sigma}\right)^2\right)$$ It is obvious that, ...
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Does IQR method for outliers work for non-normal data?

Any observations that are more than 1.5 IQR below Q1 or more than 1.5 IQR above Q3 are considered outliers. However does this theory still hold when a data set is not normally distributed? Outlier ...
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Why does Anderson-Darling statistic increase as Frequency is increased for my data?

I have been carrying out sieve analysis to estimate the median particle size of a powder. I wanted to check if the data so far followed a normal or lognormal distribution, as I have come across ...
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Is there a UMP test for testing the equality of variance between two normal groups with known mean?

Consider the case that two groups are known to be normal with the same sample size, the population mean is known. For simplicity, we can demean the sample first, say $(X_{i})_{i=1}^{n}\sim_{i.i.d.}\...
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How to combine multiple reliability data with non-equal means and normal distribution?

Suppose I have a reliability data of an item from three sources (see picture). All of the sources provide reliability with different means ($\mu_1$ , $\mu_2$, $\mu_3$) and normal distribution ($\...
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Conditional Distribution of Normally distributed random variable

Let $x \sim \mathcal{N}\left(0,\sigma^2\right)$ and $y = x+\epsilon$ where $\epsilon\sim \mathcal{N}\left(0,\sigma^2_\epsilon\right)$ and independent of $x$. We know that the conditional distribution ...
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Probability that a > b when a = b + constant [closed]

X is a random variable with normal distribution of average 260 and sd 50 What is the probability that a >= b when a = b + 15 Brand X computer have a average battery duration of 260 minutes with a ...
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Joint normality of a vector derived from joint normal vectors

Suppose that we have two random vectors following joint normal distributions: $$X=[x_1,x_2]'\sim N(0,\Sigma_X)\quad \textrm{and}\quad Y=[y_1,y_2,y_3]'\sim N(0,\Sigma_Y).$$ In this setup, I am ...
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Closed form credible region/marginal credible interval for Gaussian distribution

Suppose we have a multivariate Gaussian distribution $X \sim \mathcal{N}(\mu,\Sigma)$ where $\mu=(\mu_1,\mu_2,\dots,\mu_n) \in \mathbb{R}^n$ (the posterior of a conjugate Gaussian prior perhaps). Is ...
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When to model assuming a Poisson vs when to model assuming Normal?

I understand that If I add an infinite number of variables the limiting standardised distribution is standard normal. This comes from CLT I also understand that the summation of independent Poissons ...
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Bayesian experiment analysis of multi-cell experiment

I have an experiment running with four cells divided by two axes; the axes are (1) experiment group (test, control) and demand period (high, low). There are four total combinations therein, test-high, ...
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Joint Normality of a Random Vector with elements from a Joint Normal Distribution

Suppose that $X=[X_1,X_2,X_3,X_4]\sim N(0, \Sigma)$ (i.e. $X$ follows a joint normal distribution). Define $Y=X_1+X_2$ and $Z=X_3+X_4$. Here, as far as I know, each of $Y$ and $Z$ is normally ...
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Why the intermediate steps of a diffusion model (q(xt|xt-1)) can be considered as a Conditional Gaussian Distribution?

I have a similar doubt to this question: Why can de-noising diffusion models be sampled with Gaussian distributions? As asked in the original question, when we start from xo (the original image) and ...
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Fast top-n from samples of many different normal distributions

Let's say I have 1 million normal distributions, each with a different mean and stdev. I want to sample from each distribution and take the top 10 samples (for a Thompson sampling application.) Is ...
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Why is my data not normally distributed while I have an almost perfect QQ plot and histogram?

I am doing a thesis and I am stuck with my statistics on time series data. And when I run a Shapiro test, or any test, it says: ...
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Generating a truncated normal distribution

I want to write code, say in Python to generate a truncated normal distributed random variable on the interval $[a,b]$. I have a standard function, call it N which will generate a normally distributed ...
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1 answer
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Simulating "Realistic Data" for Statistical Problems [closed]

My friend is working on his Sociology Thesis and is currently waiting for the researchers to finish running their experiments and collecting the data that will be used in his project (socio-...
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Is normal distribution better than skewed distribution for machine learning input features? [closed]

Distribution of a particular feature in my ML dataset is skewed as shown below. Log of this feature looks like a normal distribution. Can the latter distribution offer better predictability in a ...
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3 votes
2 answers
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Distribution of an RBF-transformed normal variable

My question might be related to this or this one, but I have reasons to hope my problem is more benign. Assume I have a normally distributed variable $X \sim N(0, 1)$. What can be said of $Y = \exp(-\...
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Finding the conditional distribution from given normal distributions using Bayes' theorem

Background This question is related to my previous question: Describing the measurement of a random variable as another random variable, but I've narrowed and clarified my question. I think I've ...
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Probability Distributions And Certainty

How does certainty effect probability distributions? Here is an example: I compute the mean and variance of the weight for 10 apples and 1000 oranges. I am significantly more certain that the mean and ...
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P-value between expectation and observation of a Poisson process

Probably this is a question with a simple answer, but I was not able to wrap my head around it. I want to determine the compatibility of observing a number of events $N_{obs}$, under the hypothesis ...
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Gaussian distribution, Mahlanobis distance, probability [closed]

**Suppose we have a set of b variables which have x and y coordinates (bx1 by1, bx2 by2,...,bxN byN) each of which have a gaussian distribution. we also have somepoints, suppose (Ux1 Uy1, Ux2 Uy2, ......
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How to obtain an equivalent standard normal variate for a discrete distribution?

Let's say we have a continuous random variable with distribution function $F$. If we want to obtain a corresponding normal variate at $x$, then we simply compute $ \Phi ^ {-1 } \left(F \left( x \...
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Describing the measurement of a random variable as another random variable

Background Suppose we have a box of resistors. The manufacturer rates these resistors at 100 ohms, but they have some variability. Let $x$ be the true resistance of a resistor chosen from the box at ...
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The distribution function for for the sum of variables which follow a multivariate normal? [duplicate]

Suppose we have two random variables that follow a multivariate normal distribution, $[x,y]\sim MVN([a,b], \begin{bmatrix} \rho_1 & \rho_3 \\ \rho_3 & \rho_2 \end{bmatrix})$ Then what is the ...
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Expectation of the product of multiple correlated 1-D normal variables [duplicate]

First question: is it possible to have a set of $k$ random variables $\left\{X_i\right\}$ s.t. each $X_i \sim N(0,1)$ individually, and $\text{Corr}(X_i,X_j)=\rho$, $∀i\neq j$? If those conditions are ...
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1 answer
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Two not normal distributed samples & unequal sample size

I got two sample sets a training set around 32k rows and I have a test set of 16k rows. I want to test if the two sets are randomly split. I checked the normal distribution and it is not met for none ...
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Ensuring Correlation Between Groups of Variables

Correlation is usually defined between 2 variables - for example, height and weight measurements in athletes might be correlated, or height and salary might also be correlated. Suppose I want to ...
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Extracting statistical parameters from a mixture of two distributions of different kind

I have a dataset b (as a list in Python) of length 100 I know that is amounts to the mixture of two distributions: A normal distribution A uniform distribution ...
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2 votes
2 answers
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Similarity between closed-form and sampling-based distribution

Are there any well-known techniques that are capable of estimating the similarity between a closed-form distribution (specifically a Gaussian if that helps) and a distribution obtained by sampling? I'...
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Finding likelihood for an individual observation with cumulative distribution function of a normal distribution [duplicate]

I have to write a probit model, which will describe the effect of an education program on a grade from an exam. The data is the following: GPA: grade point average TUCE: test score PSI: ...
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Simulating Correlated Categorical Data

I am trying to simulate "correlated categorical data". For instance, consider the following example: Suppose there are 10 players (p1, p2, ...p10) - each day, a random combination of these ...
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6 votes
1 answer
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MLE for a modified German tank problem

Suppose I have a distribution $U(0,a)$ where $a$ is unknown and we are interested to estimate it. Someone who has access to $n$ samples $\mu_i$ of this distribution instead decides to create a ...
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How to deal with highly right skewed independent variable in Linear Mixed Modelling?

I am performing Linear Mixed Modelling with several independent variables. One of the independent variables is not normally distributed and contains a lot of 0`s. Around 2200 of 2555 observations have ...
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2 votes
2 answers
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What does asymptotic efficiency mean in statistic?

I reads some comparison articles, and always find " asymptotic efficiency", "asympototically less efficient", and "asympotoically normal". I really confused about the ...
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How to rearrange exponent terms in a Gaussian likelihood?

I'm working out of the textbook "Bayesian Data Analysis for the Behavioral and Neural Sciences" by Todd Hudson, and on p. 105 (above) we see the preceding explanation for a Gaussian ...
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8 votes
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Why does a zero entry in the inverse covariance matrix of a joint Gaussian distribution imply conditional independence?

When $X_1, X_2, \ldots, X_n$ are random variables jointly following a Gaussian distribution, let $A$ be the inverse of the covariance matrix ($A=\Sigma^{-1}$). I am wondering how to prove following ...
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3 votes
1 answer
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Sampling Distribution of Reciprocal of Sample Mean

Given $X_1, X_2,..., X_n $ i.i.d. random variables. $E[X_i] = \mu_1 \in \mathbb{R} $ $\&$ $ V[X_i] = \sigma_1^2 \in \mathbb{R}^+$ $\forall i \in \{1,2,3,...,n\}$. The statistics $\bar{X} = \frac{...
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Independence of first and second sample moments (about zero) of normally distributed random variables

If $X_1,...,X_n$ are independent normally distributed random variables with means $\mu_i$ and variances $\sigma_{i}^2$ and, \begin{align} M_1^2 &=\left(\dfrac{1}{n}\sum\limits_{i=1}^n X_i\right)^2 ...
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Using Box-Cox transformed features as input decreased the $R^2$ score of a regression model

I am working on building a regression model to predict housing sales price using house features (Ames housing dataset). And I prepared feature set in two ways Case 1. I performed boxcox ...
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Is it fine to present Median and IQR for normally distributed data?

I have data with more than 25 variables. Some of them are normally distributed and others are not. Instead of checking each variables for normal distribution and presenting ...
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In standard normal distribution table how i deal with value of Z greater than or equal to 5

How can i deal with sample distribution table when Z is greater or equal to 5? For example:(-2.04 <z<-5.96) Sample distribution table value for -2.04 will be 0.2018 then what will be the sample ...
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Margin Distribution N-dimensional random vector

I have this exercise on my textbook and I can't understand how I have to do it. I tried to compute the posterior but I don't understand if it's what the exercise requires. Thanks This is what I did ...
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Integral of the product of two gaussian

Here is the big picture of my problem: In the image below, X and Y represent 2 independent gaussian distributions. So the circles are the representation of the resultant bivariate gaussian. This ...
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What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?

Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of $$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
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Estimate normal distribution parameters from smallest N samples

I have a bunch of small datasets (billions of sets of 7 samples). Each dataset represents the smallest 7 samples of a larger set of 15 values which are normally distributed. Given just the smallest 7 ...
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Wasserstein distance between multivariate lognormal distributions

Wikipedia gives the following formula for normal distributions: What changes, if any, do I need to make to handle multivariate lognormal distributions instead?
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For bivariate linear model, if the residuals ARE normal, but the data is NOT normal, can I make inferences about the slope coefficient? [duplicate]

For convenience and context, I'm looking at the formula for standard error of the slope coefficient from here: https://www.statology.org/standard-error-of-regression-slope/ If X is not normally ...
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Testing for constant mean (stationarity)

I have a grayscale digital image with noise and I would like to test for uniformity in small neighborhoods. One hypothesis says that the mean is independent of the spatial coordinates, against the ...
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