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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Confidence interval for normals of normal model

Suppose $X$ is a normally distributed random variable with unknown mean $\theta$ and known variance $\sigma_{X}^{2}$. Suppose that $Y_{1}, \ldots, Y_{n}$ are random variables that, conditioned on $X$, ...
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Arithmetic with normally-distributed variables

Suppose $x \sim \mathcal N(\mu_x,\sigma_x^2)$ and $y \sim \mathcal N(\mu_y,\sigma_y^2)$ are random variables, and suppose $\mu_y$ is large compared to $\sigma_y$. I want to know about $$ z=\frac{x}{y^...
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Why use normality tests if we have goodness-of-fit tests?

What are the reason/s to use a normality test (e.gr., Shapiro-Wilk, Jarque-Bera) instead of a goodness-of-fit test to a normal distribution for some data we want to check for normality?
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Normality Assumption - Help! [duplicate]

I am relatively new to statistics and struggle with the normality assumption (where and how it needs to be assessed). I understand that parametric tests need the data to be normally distributed. The ...
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What is the distribution of the daily change rates of S&P 500?

I was wondering what is the distribution of the daily change rates of the S&P 500. It doesn't seem to be distributed normally (using qq plot for example).
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Showing that the given critical value (?) gives a test with significance level 0.05

So I have a maths problem that I'm struggling to understand... The original language isn't English, but I have done my best to translate the necessary background information into English. Assume that ...
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Probability that interval arithmetic estimate of mean covers the true mean of a normal variable?

Background Suppose there exists a normal variable $X \sim \mathcal{N}(\mu, \sigma)$ where $\mu, \sigma$ are known. Also suppose that you only observe rounded instances $Y = \operatorname{round}(X, d)$ ...
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Distribution of MLE of variance of the normal distribution [duplicate]

The MLE(not the unbiased one) of the variance of the normal distribution is $$\ S^{2}=\frac{\sum_{j=1}^{n} (Y_j-\bar{Y})^{2}}{n}$$ I am trying to find its distribution.Can anyone help?
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what is the formula for confidence interval of standard deviation of a sample where observations are proportions of people from different countries? [closed]

example of a sample. What is the formula for confidence interval of standard deviation of people, having an access to sanitation facilities, if taking this sample into account?
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Mean of skew normal distribution with normal prior obtained with Gibbs sampling

I would like to obtain a new mean $\mu$ of a skew normal distribution with a normal prior of the form $N(\delta,\tau)$ on $\mu$, and a given standard deviation $\sigma$ and shape parameter $\alpha$. ...
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Inequality related to normal distribution

Prove $$\exp(\frac{z_{\alpha/2}^2}{2}) \leq \frac{2}{\alpha}$$ where $z_{\alpha/2} \sim \Phi^{-1}(1-\frac{\alpha}{2})$ Does this inequality hold? How to prove it? Thanks!
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What Explains the Behavior of these Graphs?

I thought of the following experiment: Suppose we generate a single random number from a normal distribution (with a specific mean and specific standard deviation), we then take the difference of this ...
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How does a patients distribution affect a trials distribution?

For example, the trial has a normal distribution with mean 10 and stdev 35. However, that distribution is calculated using the average of each patients measurements. Say each patient in the trial has ...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

Note: I also posted this question on MATHEMATICS. For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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Fitting discrete data to continuous distributions

I'm creating a simulation model, in which some stochastic factors are included. On of my stochastic factors is the amount of containers arriving daily for a specific delivery location. A plot of this ...
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Distribution of true value when measurement imprecision is non-constant

What I believe to have understood so far (I am not a mathematician or a statistician, so please correct me if I'm wrong.) Say we are making measurements of some phenomenon $X$, and we have a normally ...
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QQ Plot help meaning [duplicate]

How can I interpret the following QQ Plot? Can you explain it for example for the point 20 and 12?
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Is sampling from $\mathcal{N}(\mu, \sigma)$ equal to sampling from $\mathcal{N}(0, 1) * \sigma + \mu$? [duplicate]

This is a simple method to transform samples from $\mathcal{N}(0, 1)$ into samples from $\mathcal{N}(\mu, \sigma)$ with arbitrary $(\mu, \sigma)$, without having to re-sample from $\mathcal{N}(\mu, \...
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The derivation of the RBF kernel to the inner product form, and what does this notation $\sum _{n_{1}+n_{2}+\dots +n_{k}=j}$ mean?

I have two questions regarding the following derivation: What does this notation $\sum _{n_{1}+n_{2}+\dots +n_{k}=j}$ mean in the following equation? How is it derived from step 3 to step 4, and from ...
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Normalize a Dataset with Min/Max formula [closed]

I have 7 groups of data (let's say g0 all the way to g6) I want to normalize g0, and I apply the following function only to the elements of g0: ...
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Probability for all points on a path in a gaussian random field

I have a gaussian random field $u(x)$, $x \in R^2$ , with a covariance function $C(d)$ and I need to calculate probability $P[ u(x) < u_0, \forall x \in S]$, where $S$ is a straight segment in $R^2$...
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Range for Euclidean distance between two variables [closed]

I have two datasets A and B. I want to compare elements of those datasets to find "matches", i.e., elements from ...
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Can you help me with distribution/Skewness

Can you tell me if the data normal distributed are? It looks a bit strange. I would say it is skewed 40,80 and so on are sizes of the Apartments in m^2
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Distribution function of the linear combination of standardized student-t quantiles [duplicate]

Assume to observe 2 quantiles, x and y, associated with the z% probability. These quantiles are generated by 2 non-independent standardized student-t distributions X and Y. In case of linear ...
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How to interpret a point in the qq plot? [duplicate]

I am trying to understand the Q–Q plot. Suppose I create a sample according a exponential distribution ...
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Spherical Normal Distribution [closed]

In the following image, the vertical direction has the probability density and it is based on two variables x1 and x2. Image Ref: Tail probabilities of multivariate normal distribution In a similar ...
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Is "Information" somehow Related to "Variance"?

Recently, I have learned about the principle of Maximum Entropy with regards to Probability Distribution (https://www.youtube.com/watch?v=2gTrsLVnp9c) - in particular, when certain "information&...
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Do Mixture Models "Defy" Entropy?

Recently, I have learned about the principle of Maximum Entropy with regards to Probability Distribution (https://www.youtube.com/watch?v=2gTrsLVnp9c) - in particular, when certain "information&...
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Normal Distribution or Not

I have some net yield data (N=700) from a survey and I intend to carry out analysis to identify if there is any significant difference or relationships between the yield and the characteristics of the ...
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Expectancy of Euclidean norm of correlated bivariate Gaussian [duplicate]

Consider a noncentral bivariate Gaussian , with nonzero mean, and with ρ not neccesarily zero, and σ1 not neccesarily equal to σ2. I want to find an expression for the expected value of the Euclidian ...
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computing the expectation of a variable

I'm trying to run a simulation but before I need to compute the exact value of X, where: V ~ Bernoulli(0.5) X|V ~ Normal(1-2 V, 1) The question is : what is the expectation of X (E(X)) in the ...
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Fitting variable-dependent normal distribution to data

Given a sample, one can usually find the best fitting normal distribution by matching the mean and variance. What's the correct way to fit a normal distribution to data when the parameters aren't ...
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Entropy of Gaussian mixture when variance of one component gets larger

I want to prove or disprove the following relation of differential entropies: Conjecture: $\displaystyle h(f) \le h(g)$ where $f, g$ the density functions of Gaussian mixture models with equal ...
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Comparison of normality + similarity distribution of time series datas

I have a training time series data , whose last 20% is kept as validation data. I want to check whether the distribution of training and validation features are similiar and normal, so that we can say ...
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Estimating Mixture Models with Maximum Likelihood

Suppose you have a Normal Mixture Model with 2 Components - you could write this model as follows: $\pi_1 N(\mu_1, \sigma_1) + \pi_2 N(\mu_2, \sigma_2)$ In the above model, there are 6 unknowns : $\...
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Calculate posterior distribution given two normal distributed measurements

I have two measurements $m_x$ and $m_y$ from two sensors $X$ and $Y$. $m_x$ can be approximated by $m_x \sim \mathcal{N}(m_x|m, 2)$ and $m_y$ can be approximated by $m_y \sim \mathcal{N}(m_y|3m, 5)$. ...
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Unexpected distribution of ab-cd where a,b,c,d are independent and N(0,1) distributed

I have discovered an unexpected curiosity that if a,b,c,d are independent random variables and are N(0,1) distributed, then |ab-cd| is exponentially distributed Exp(1). It's easy to verify that ...
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What statistical analysis should I use [duplicate]

I have been measuring flux of CO2 in (mmol m-2 d-1) from two different rivers and I now want to do a statistical analysis using IBM SPSS but Im not sure where to start. One lake is showing negative ...
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Is there a way to determine possible errors in data based on distribution

I am trying to find possible errors in the data. Below are the distributions of yield for 4 different fields. As the fields are large and non-homogenous, the distribution is not normal. Previously, I ...
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Bayesian update with the shifted and scaled data

Suppose I have data $y$ (N observations) which follows a normal distribution: $y \sim N(\alpha+\beta*\mu,\sigma^2)$ while $\alpha$ and $\beta$ are known parameters. I want to update $\mu$ and the ...
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Intuition for why mean of lognormal distribution depends on variance of normally distributed rv

Let $X\sim\mathcal{N}(\mu,\sigma^2)$, which is a normal distribution. Then, $\text{exp}(X)\sim\text{Lognormal}(\mu,\sigma^2)$, and its mean is $$ \mathbb{E}[\text{exp}(X)]=\text{exp}\left(\mu+\dfrac{\...
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How general is this property about correlation and the sum of two normal RVs?

(Cross-posted from math stack exchange as I didn't get any responses there) Given a random vector $(X_1,X_2)$ that is jointly normal with means / sd's $\mu_1,\mu_2, \sigma_1,\sigma_2$ and correlation ...
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Statistical analysis of carbon dioxide flux data

I have been measuring flux of CO2 in (mmol m-2 d-1) from two different rivers and I now want to do a statistical analysis using IBM SPSS but I'm not sure where to start. One lake is showing negative ...
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1 vote
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Asset prices are in theory log-normally distributed

In finance, the price of an asset is given by the following formula. $P_t=P_0* e^r$ r=returns Pt = Price at time "t". P0 = Actual Price. Likewise, it is assumed that the yields (r) will ...
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Simplifying the Kullback-Leibler divergence for a sum of distributions

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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Density of Multivariate Normal Distribution (MVN): Dimension = 1 or k?

I have a very naive question about the density function of the MVN distribution. According to the wiki page, the density function formula sometimes has a constant k ...
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How is the difference between 2 normal distributions a normal distribution? [duplicate]

I've learned that if we sum or subtract 2 normal distributions, the result would be another normal distribution with $\mu=\mu_{1}\pm\mu_{2}$ and $\sigma=\sqrt{\sigma_{1}^2+\sigma_{2}^2}$ https://www....
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Variational Inference Mean-Field Gaussian

I am new to variational inference and got very confused about some basic ideas. We want to use the mean-field gaussian family to approximate a complicated high-dimensional distribution. I want to ...
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Distribution of Residuals and Response - When to use which?

I would like to analysze the relation between my continuous response 'soil moisture content [%]' and 2 categorical and 1 continuous predictors. I fit a linear mixed model and checked the distribution ...
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Is return normal or log normal?

I read that return is normal and stock price is log normal. But I also read that return is log normal. So I am confused about which it is. In the 14th Chapter of Options, Futures, and other ...
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