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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Determine confidence level in guesstimate: gas stations in Germany

I read in the internet about the following guesstimate during a job interview for a trading position: Give an estimate for the bid-ask on the number of gas stations in Germany (with max spread of 10 %)...
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What are the values of the draws from the uniform distribution used to create x? [closed]

Any help is greatly appreciated. Please let me know if there is any other information that is needed.
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how to estimate parameters of a normal-gamma distribution from a data sample?

The short version is: how to estimate alpha, beta, kappa and mu of a normal-gamma distribution from a data sample? I've been trying to use this implementation for bayesian inference: https://github....
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What technique to use for non-normally distributed multiple measures data (continuous outcome)?

I have repeated measures data of size around 100 sample, in which a biomarker is measured at different time points. The patients are randomly divided into Placebo and DrugX group. But the outcome (...
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Computing KL divergence between uniform and multivariate Gaussian

Another post has addressed the fact that KL divergence is defined between a uniform distribution and a Gaussian distribution $$D_{\text{KL}}(\mathcal{U}(x) \parallel \mathcal{N}(x \mid \mu, \Sigma)) = ...
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Estimating confidence interval of a parameter from the MLE of another parameter

Let's say I have a maximum likelihood estimate (MLE) for my parameter $\theta$ as in $$\hat{\theta}\sim N\left(\theta, \frac{\theta^2}{n}\right) $$ and we have $$\mu = \sqrt{\frac{\theta}{2}}$$ If I ...
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Cholesterol levels that follow Normal Distribution and 2 simple questions [closed]

Let cholesterol levels of a population be described by a normal distribution $X \sim N(μ=250, σ = 50), \quad P_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-(x-250)^2}{2 \times 50^{2}}}$ I am asked to compute ...
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experiment design: effectivelly sample the individuals while having cost constraints [closed]

First, i already asked a relatively similar question here. Sorry, if it seems to be a duplicate. we are constructing a study were subjects (people) have to perform a task. While doing it, we measure a ...
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What is the difference between normal distribution and standard normal distribution?

While studying statistics I am stuck with a question of 5 difference between normal and standard normal distribution. Standard normal dist. has mean 0 and standard deviation 1 and for normal dist. it ...
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63 views

How to prove the following properties about $\chi_1^2$?

Given two independent r.v. $X\sim N(\mu_1, 1)$ and $Y\sim N(\mu_2, 1)$. What is $$ f(x)=P(\min\{X^2, Y^2\}>c). $$
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what is the probability of sample variance when true variance and true mean is unknown?

Sample Variance by definition is $s^2 =\frac{1}{n-1} \sum{(x_i-\bar{x})^2}$ When the population distribution is normal and true variance $\sigma^2$ is known, Sample Variance follows the chisq ...
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Probability that the absolute value of a normal distribution is greater than another

Greatly appreciate anyone that is willing to Help. I am thinking about the question of comparing the absolute value of normal distributions. Given $a > b$, $X$ ~ $N(0,a)$ and $Y$ ~ $N(0,b)$, what ...
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Why is the cholesky decomposition the correct matrix square root to sample from? [duplicate]

I recently discovered that not all matrix square roots are the same because they are basically rotations of one another. Assuming that $X$ is already mean centered, we could use some different ...
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Hypothesis Test for mean vector of Multivariate Normal Distribution

Given two independently $X_1$ and $X_2$ and that these are bivariately normally distributed with mean vector components $\mu_1$ and $\mu_2$ and variance-covariance matrix shown below: If we want to ...
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Maximum Likelihood Estimators of Multivariate Gaussian as $\mu=[\mu_1, \mu_2]^T$? [closed]

As in Maximum Likelihood Estimators - Multivariate Gaussian, Assume that we have $m$ random vectors, each of size $p$: $\mathbf{X^{(1)}, X^{(2)},...,X^{(m)}}$ where each random vectors can be ...
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Spearman correlation test and linear relationship vs monotonic relationship?

I want to use Spearman's correlation test. My data is not normally distributed. Below is a scatterplot of this data. I read somewhere that Spearman's correlation coefficient can describe monotonic ...
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Risks for the Coefficient of Variation when using very skewed (log-transformed) data

I have many datasets which I have normalized using log-transformation, however for some datasets the log-transformation did not improve the skewness that good (still close to 1). For these datasets, I ...
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MAD & Median of weighted GMM

What is the median and median-absolute-deviation of a weighted GMM in terms of component mean and variance? For example, three normal distributions $A$, $B$, $C$ with means $\mu_a,\mu_b,\mu_c$, ...
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Integrating out variables in a Gaussian copula density

Consider a multivariate continuous distribution with a Gaussian copula, i.e. we can write its PDF as $$ p(x)= \biggl(\prod_{j=1}^D p_j(x_j)\biggr) c\bigl(F_1 (x_1), \dots , F_D(x_D)\bigr) \enspace, $...
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non gaussian residual distribution for ARIMA

I have a few theoretical and practical questions regarding residual distribution of ARIMA models. I made quite a few for various purposes and I don't think I ever found gaussian distributed residuals. ...
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Normal Distribution - Implications of negative values

Consider a task, whereby we need to generate a normally distributed data set of 100 numbers. "tableFreq" is being used to hold the "frequency" data. The sum of the frequencies ...
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Can any covariance factorization $LL^\top$ be used for sampling?

I thought that any factorization of the for $LL^\top$ of a covariance matrix could be used for correlating random noise according to the covariance. I tried doing this with the following code and ...
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If $x$ is a normally distributed random variable, then what is the distribution of $x^4$ ? Does it follow a well-known distribution?

I am interested in the PDF of $x^4$ if $x$ is a normal distributed variable with non-zero mean. The question is related to this post that considers the cubic of a normal distributed variable. However ...
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Why are MLE for high dimensional multivariate gausian covariance matrix likely to be ill-conditioned

In a book I'm reading (Probabilistic Machine Learning: An Introduction) the author suggested that in high dimensions, the MLE estimate for the covariance matrix for multivariate gaussian is often ...
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Normalization in curve fitting

I'm doing a stat course and I don't understand this at all. Would really appreciate if someone could explain under what circumstance the fitting function is supposed to be a PDF and when it should ...
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Is a left-side truncated distribution similar to a Poisson distribution? [closed]

I'd like to find a proper way to deal with the distribution of the sum of one-side truncated normal distributions. And I notice that Poisson distribution is similar to a truncated normal distribution (...
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Averages of the two closest pairs out of a set of four observations

Four random numbers are drawn at random from a standard normal distribution. They are grouped in two pairs of closest numbers, $\{x_1, x_2\}$ and $\{x_3, x_4\}$ so that $x_1\le x_2 \le x_3 \le x_4$, ...
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Gradient of Mixture of Normals in log scale [migrated]

I have a mixture of two normal distributions with mixture parameter $$ \pi(x) = \alpha \mathcal{N}(x; \mu_1, \Sigma_1) + (1-\alpha)\mathcal{N}(x; \mu_2, \Sigma_2) $$ What is the gradient of the log ...
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How to show that this integral of the normal distribution is finite?

Numerically, I have noticed that $$\int_{-\infty}^{\infty} \dfrac{\phi(x)^2}{\Phi(x)}dx < \infty$$ where $\phi$ and $\Phi$ are the standard normal pdf and cdf. However, I do not see how to prove it....
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Distribution of the sum of random variable bessel and the normal distribution? [closed]

What is the distribution of the sum of two random variables one is Bessel and the other is normal?
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Size of a sample needed for good estimation of multivariate normal distribution [duplicate]

I'm currently dealing with a problem where an underlying assumption is that, in general, subsets of my data follow somehow distinguishable multivariate normal distributions. I'm estimating via MLE ...
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Independent increments in a Gaussian Process

Sorry if this is a naïve question, but if you have a gaussian process: $$ X = \{X(t), t\ge0 \},\ X(t) \sim \mathcal{N}(0,t) $$ Can you prove that it has independent increments? If yes how? And if no, ...
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Comparing outliers in two distributions

I apologize in advance as I am not well-versed in statistics, but I hope that this question makes sense. I have 2 populations which are normally distributed and have a near-identical mean. I would ...
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Honest trees giving asymptotically Gaussian estimates

I am trying to fully understand the ``Estimation and Inference of Heterogeneous Treatment Effects using Random Forests" paper by Wager and Athey. In this paper it is stated that using honest ...
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1answer
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Is the idea of "Zeno's Paradox" indirectly assumed in every statistical model?

I have more of a philosophical question : Is the idea of "Zeno's Paradox" indirectly assumed in every statistical model? In philosophy, there is an old concept first attributed to the ...
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Distribution for the number of Gaussian random variables in a given bin (or interval)?

Suppose I have N Gaussian random variables (assume 1d) with mean $x_n$ and standard deviation $\sigma_{x,n}$. Suppose I then generate samples from these random variables, and bin the generated data ...
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Shapiro-Wilk test on 2 samples Python

I'm checking if the samples are normally distributed. In samples are two deep learning models losses calculated during experiments. As I know based on Shapiro-Wilk test formula, we are checking if ...
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Why logit transformed binomial proportion is approximately normal?

What is the argument to prove asymptotic normality of logit transformed of binomial proportion which follows Beta distribution? $\theta$ has beta prior and model $y|\theta$ follows $Binom(\theta,n)$. ...
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Quartic From Gaussian Expectation

$x$ is $d$-dimensional vector, Gaussian distributed with mean $\mu$ and covariance $\Sigma$. I want to simplify the same expression attached below but replacing $x$ with $x^2$ in the beginning of the ...
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A mixture of discrete and continuous components

Suppose that random variable $X$ is sampled from Bernoulli with probability $\pi$. Let $Y\sim N(\mu, \sigma^2)$ and denote $Z=XY$. Then, when $X=0$, $Z$ equals 0 (and this happens with probability $1-\...
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Does it make sense to use t-student distribution for stock returns monte carlo simulation?

If we assume that stock returns are normally distributed and we want to download some of the data of a stock, then we will practically always take only a small part of it (we will not take for example ...
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Why are residuals required to be Gaussian in linear regression?

Let us consider two column vectors of random variables ${\bf Y} = (Y_1,\ldots,Y_n)^{\intercal}$ and ${\bf X} = (X_1,\ldots,X_m)^{\intercal}$. The general linear regression model is written as \begin{...
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Expectation involving i.i.d. complex Gaussian random vectors

$\boldsymbol{h_1}$ and $\boldsymbol{h_2}$ are i.i.d. circularly symmmetric complex Gaussian random vectors with zero mean and covariance matrix $\boldsymbol{K}$. $ \boldsymbol{h_1} = \left [h_1(0),\...
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Statistical analysis of distributed values in Java [closed]

I am writing a program in Java that outputs a List<Double> of distances that roughly follow a bell curve distribution. From this data, I need to generate two ...
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Expectation of square root of scaled noncentral chi-squared minus a constant

Suppose that $X\sim \mathcal{N}(\mu,\sigma^2)$ and $\mu>\sqrt{c}$. Is there an approximate closed expression for $\mathbb{E}[\sqrt{X^2-c}]$? My application allows me to impose an upper bound on the ...
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Concept Clarification: Wilcoxon rank test vs two-samples T-test (UPDATE)

I am currently deciding whether to use the Wilcoxon rank test vs two-samples T-test to assess BMI between two unrelated groups. Using the following code to assess normality ...
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1answer
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three sigma vs six sigma

I wonder about the meaning of the term six sigma. It describes a process/production with (mostly) zero defects. You can see here: source: https://www.dummies.com/article/business-careers-money/...
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integration of product of a gaussian pdf and a student-t pdf

I want to perform the following integration wrt $x$: $$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$ Here first part is ...
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Why the null hypothesis for AD is rejected for sample with positive skewness?

I know that the AD test for normality is very sensitive at the tail, but I notice that the null hypothesis for AD is rejected for samples with positive skewness, whereas it is not rejected for samples ...
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Convergence of Gaussian random variables

Let $(f_n)$ be a sequence of 0-mean Gaussian densities on $\mathbb{R}^d$ and assume $f$ is limit of $(f_n)$. Question 1 How does one determine the type of convergence by looking at the corresponding ...

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