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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Estimating a secret with before/after interchanging noises
$\newcommand{\Var}{\mathrm{Var}}\newcommand{\E}{\mathrm{E}}$
For $n+1$ iid "noise" variables $X_0,\dots,X_n$ from the normal distribution $\mathcal{N}(0,1)$ and a "secret" $s$ ...
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Conditioning on two normal variables $E[x|y=y_0,x\leq k]$ and obtaining an analytical expression
suppose $x=a+e$ and $y=b+e$, where $e$ is normally distributed with mean 0 and variance $\sigma^2_e$. $a$ and $b$ are independent and normally distributed, both with mean $\mu$ and variance $\sigma^2$....
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Is my explanation of how MLE works is correct? [duplicate]
The likelihood of observing the dataset we have (for some mean and variance) can be written as the product of the likelihood of observing each data point (since all the rows are independent). Now, ...
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Can I assume normal distribution?
I have calculated daily price returns of Bitcoin and plotted this data in the following way:
x-axis: returns in %
y-axis: count
I assumed the data had a normal distribution to calculate the
$$\rm ...
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All uncorrelated marginals are independent: Only for joint Gaussian?
Let $X$ be a random vector in $\mathbb{R}^p$, where $p\geq 2$, with the following property: Any two uncorrelated marginals are independent. Formally:
(1) For any $\alpha,\beta\in \mathbb{R}^p$, if $...
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Combining factors, represented as normal distributions, to one combined factor, normally distributed
I'm trying to combine the different factors that may affect running pace, such as GPS-measured distance, grade, terrain, heat and other factors (such as wind etc.). Each factor is represented as a ...
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Will this converge to origin?
Suppose you have a diffusion of 100 points with the following iteration:
$$(x_{n+1},y_{n+1}) \sim \mathcal{N}\left((x_n,y_n), \frac{x_n^2 + y_n^2}{2} I_{2 \times2}\right)$$
This will make a high ...
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Variance of powers of a standard normal random variable
To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
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The Tsallis entropy of generalized Gaussian distribution
I would like to discuss the computation of the Tsallis entropy for the generalized Gaussian distribution. From the paper in the link https://www.sciencedirect.com/science/article/pii/S0167947322000822....
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What is the variance on the ratio of dependent random variables
I have the data on thousands of emission lines such as the one shown in the figure below. A single emission line covers $N$ pixels (11 in this example). Because the data come from counting photons, $...
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How to calculate the likelihood for a normal distribution N(theta, 1) if we only know the maximum of a sample?
Assuming iid samples x ~ N(theta, 1), we have a sample of 5 observations with maximum value = 3. How to calculate the likelihood?
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Unifying multiple sample data (meta-analysis)
I would like to unify multiple sample data (of different sizes) into some "unified sample" to evaluate its collective variance. Is this something statisticians do? I thought of unifying ...
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Predictive Distribution in Gaussian Process for Machine Learning
I am reading Gaussian Process for Machine Learning equation 2.9, where it is deriving the predictive distribution
$$p(f_* | \mathbf{x}_*, X, \mathbf{y}) = \int p(f_* | \mathbf{x}_*, \mathbf{w}) p(\...
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In a skewed sample with a large n, does Central Limit Theorem dictate that a t-test can be used, even if the mean cannot be interpreted? [duplicate]
I understand that, in the case of a highly skewed population and sample, the sampling distribution of the mean can still be normally distributed if the sample size is large, according to Central Limit ...
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Why is it the convention to take equal tails in a two-tailed test with a statistic following a symmetric distribution?
Is there a particular reason for conventionally dividing the tails equally in a two-tailed test? Consider an $\alpha$ level test with a statistic following standard normal distribution. Then, why do ...
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Distribution looks roughly normal on a q-q plot, but has a p-value of 0.0 for the Shapiro-Wilk normality test. How to interpret? [duplicate]
The distribution is as follows:
However the Shapiro-Wilk test yields a p-value of 0.0 and a W statistic of 0.9. There are over 7,000 values in the sample.
Note, the quantile values have been ...
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Expectation of a normal variable given that a signal is below a certain value
If $\tilde{u}$ is a normally distributed random variable with mean $q$ and variance $\sigma$, and $s=\tilde{u}+\tilde{e}$ where $e$ is also normally distributed with mean 0 and variance $\sigma_s$. $\...
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How to determine significance for a Corrado rank test in an event study?
I apologise if this is a stupid question (it feels stupid tbh). I am currently doing an event study and my abnormal returns are not normally distributed. I am now in the process of performing a ...
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Understanding usage of quantile function (norm.ppf), passing p vs 1-p
I was given a question related to the quantile function using scipy.stats.norm.ppf, along with its solution - which I don't understand.
I'm changing the question ...
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Why does the reparameterization trick work when some components are still stochastic? [duplicate]
I am trying to understand the reparameterization trick. I got some intuition while looking at this popular question, but I still feel largely confused. I am putting my understanding and doubts here ...
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Why do Poisson regression and Linear regression give the same error? [closed]
I use the badhealth data as an example, I'm modelling the number of visit (doctor) based on the health condition and the age as two features. Mean absolute error is ...
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Finding the Variance of the MLE Variance of a Joint Normal Distribution
I have a random sampling of $Z_1,...Z_n$ from a normal distribution $N(\mu,\sigma^{2})$. I am considering them within a joint likelihood function.
I know that the MLE ($\hat\sigma^{2}$) of $\sigma^{2}$...
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Deriving covariance of joint distributions of MVN [Linear Gaussian systems?]
Let $z$ ∈ R^L be an unknown vector of values, and $y$ ∈ R^D be some noisy measurement of z. We assume these variables are related by the following joint distribution
$p(z) \sim N(z|\mu_{z}, \Sigma_{z})...
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What is the substantive meaning of one statistical test is more powerful than another?
There are some research claims that one statistical test is more powerful than another. For example, a highly cited study states:
Results show that the Shapiro-Wilk test is the most powerful ...
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Interpretation of Anderson–Darling test
Assume that I am using Anderson–Darling test to evaluate whether a given sample of data is drawn from a normal distribution with some mean and standard deviation.
If you accept the null hypothesis in ...
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Computing an integral that reduces to $\mathbb{P}[X>Y]$
Problem
Evaluate $$I=\int_{-\infty}^\infty \frac{e^{-\frac{1}{2}\left(\frac{x-\mu)}{\sigma} \right)^2}}{\sigma \sqrt{2 \pi}}\frac{1}{1+e^{-x}}\, \mathrm{d}x$$
My attempt
Now the first part of the ...
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Fit a curve with unknown function assuming normal errors [closed]
Let say that I have a 1D experimental dataset $y(x)$, which contains noise due to experimental error. Is it possible to estimate the data by assuming that the errors are normally distributed? (The ...
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Entropic Value-at-Risk for a Normally Distributed random variable
I have been reading about the Entropic Value-at-Risk measure and its applications. The definition of this risk measure is as follows [1,2]:
$${\displaystyle {\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}...
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If I draw the variable "m" from M~N(0, 1), then draw "x" from N(m, 1), what is the distribution of "X" (not X|M)
Step 1:
I draw the observation "m" from M~N(0, 1) (i.e. a Normal distribution with mean 0 and variance 1)
Step 2:
Then I draw "x" from N(m, 1) (i.e. a Normal distribution centered ...
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Can we transform a t-distribution into a normal distribution? [closed]
We know that any normal distribution can be transformed into a t-distribution as shown in this post: Transformation of any normal distribution into a standardized t-distribution
My question is can we ...
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Product of normal multivariate distributions [duplicate]
I read a book about statistics and machine learning, and can't understand assertion that: let $P(y) \sim N(y|\mu^*, \Sigma^*)$, i.e. multivariate normal distribution $p(y_1|\mu, \Sigma)$, where $\mu_1 ...
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Normally distributed variables in t-test
I'm newcomer in statistic so I'll ask probably stupid question: if normally distributed observation required to use t-test, why couldn't be used z-statistic and subsequent calculation of probability?
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Existence of random variables whose difference of squares is normally distributed
Are there independent random variables $X$ and $Y$ such that the distribution for $Z = X^2 - Y^2$, is $Z \sim N(0, 1)$?
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Can this feature be used as input to cnn based on the q-q plot?
I have normalised my input features and target variable using the quantile transform by sklearn. The Q-Q plot of one of the feature after normalisation is given as below. Is this feature normalised? ...
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Cumulative distribution of Gaussian conditional independent random variables
Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
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Calculation of an optimal variational distribution for covariance parameters in a Bayesian graphical lasso model
Context:
I am considering here a variational Bayesian framework where I need to calculate the optimal variational distribution for some covariance parameters.
Formally the model can be expressed as:
$$...
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Bayesian inference on model parameters from summary statistics alone
Consider quantities $y_1,y_2,\dots,y_p$, for the $j$th of which we have $n_j$ measurements $y_{1j},y_{2j},\dots,y_{n_jj}$. Unfortunately, I do not have access to the raw data $y_{ij}$ -- only to the ...
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Properties of the inverse normal cdf and permutation probabilities as models for horse racing
Let $T_i$ be the running time of horse $i$ and $T_i \sim N(\theta_i,1)$ and the $T_i$'s are independent. Then Henery (1981) showed that the probability $P(T_1<T_2<\cdots <T_n)$ can be ...
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Probability of being maximum value of numbers sampled from normal distribution
There are n normal distributions with mean ordered from largest to smallest $m_1$, ... , $m_n$. The standard deviations are also different but not ordered. Let's define $p_i = Pr(x_i = max(x_1, x_2, .....
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GAM: mgcv model with kurtosis: Does this need to be solved and how?
I am trying to model CO2 fluxes (fco2) using a number of environmental parameters using a GAM in mgcv. Specifically, I have leaf temperature (tl), vapour pressure deficit (vpd), and transpiration (tr)....
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Why convolving a function with a Gaussian kernel is the same as adding a Gaussian noise to the input? [duplicate]
I am implementing accelerated Langevin Dynamics (LD) for posterior estimation with prior presented with deep autoregressive network from paper [1]. I have a question about the prior smoothing ...
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Show minimal sufficient statistic is not complete in normal distribution
Let $Z_i$ for $1 \leq i \leq n$ be a sample from the $N(ap, bp(1-p))$ density, where $a \gt 0, b \gt 0$ are known but $p \in (0,1)$ is an unknown parameter.
I have shown that $T = (\sum^n_{i = 1} Z_i, ...
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Most probable value for the successor (in ascending order) of a known statistical unit
Let $n$ be an integer $>1$.
Suppose a sample $(x_{k})_{k\in[[ 1;n]]}$ is taken from a known distribution on $\mathbb{R}$.
Given $x_1$ and supposing $\exists k\in[[2,n]], x_k>x_1$, what is the ...
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von Mises-Fisher distributions and Gauss characterization of Gaussians
We can read in Jaynes' Probability Theory: The Logic of Science that the family of Gaussian densities
$$ f_{m}(x) = \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-m)^2}{2\sigma^2}} $$
with $m\in \mathbb{...
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Papers or documents about the central limit theorem and its possible extensions: what happen when the sample size is big? [closed]
The central limit theorem in its most popular form states that (without being too formal) for a set of random variables $X_1,X_2,...,X_n$ independent and identically distributed with mean $\mu$ and ...
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Marginal Gaussian distribution
I have a confusion over an integral involving a multivariate and a univariate Gaussian.
We know that in the case of two multivariate Gaussians the following is true:
$$
\int \mathcal{N}(\mathbf{y}|\...
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Best estimator of the mean of a normal distribution based only on box-plot statistics
Suppose $X_1,\ldots,X_n\sim\operatorname N(\mu,\sigma^2)$ and you can observe only the sample size $n,$ the two extreme values, and the first, second, and third quantiles of the sample. Among unbiased ...
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Use linear mixed model or linear quantile mixed model for non-normal residuals?
I started with this initial model:
m1 <- lmer(response ~ treatment + (1|subjectID), data = data)
However, the residuals of the model are heavy-tailed (presumably enough to violate the normality ...
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Question on the normality assumption of the predictor variable and the covariance matrix of the likelihood
Suppose you are trying to fit a continuous variable y depending on some input x ( es y= ax+b) and you are using a multivariate distribution as likelihood for the fitting.
Does using a multinormal ...
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Are all examples of normal distribution in nature only a consequence of CLT? [duplicate]
I have read that many bell curves we see in nature is just a consequence of the CLT, because those things are just the result of many small additive causes (e.g. human height).
Then my question is: ...