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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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20 views

check for noncorrelation of normal pseudo random numbers using scatterplot

pseudo random number generators should give as output random sequences u1, u2, ... that are independent and identically distribuited (iid). Since testing for independence is not easy, the first check ...
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2answers
42 views

Normal Distribution mean

What does it mean when we say "consider a normal distribution in variable $x$ whose mean is a linear function $Ay+b$ of second variable $y$"? As per my understanding there is 1 mean of a normal ...
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Question about the log-normal distribution

Let's say that $Y = X + Z$, with $X$ and $Z$ being independent, and $Y$ having the same distribution as $Z^2$. In addition, $X \geq 0$ and $Y, Z \geq 1$. The latter condition is needed for reasons ...
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To converges Multivariate t dist. to Generalized Gaussian dist

The density of generalized Gaussian distribution with order parameter $s>0$ is $$ f(x) = c(m,s)\exp\left(-\frac{\|x\|^s}{s}\right), \quad x\in \mathbb{R}^m, $$ where $c(m,s) = \frac{\Gamma(m/2 + ...
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1answer
13 views

Scale parameters in a generalized normal distribution

Lets say that I have variable $X$ which follows a generalized normal distribution with parameters $(\beta, \mu, \sigma)$ and I wish to change it so that it has parameters $(\beta, \tilde{\mu}, \tilde{\...
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Which data is “more normal”?

I have two sets of data, and I want to test which is "more normal" (specifically residuals from two different models fitted to hourly and daily data - the daily data is the hourly data aggregated). ...
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if the mode of a normal distribution is 0, then what's the value of the mean

so if the PDF attains its maximum at 0 it means that $f'(0) = 0$: $$f'(x) = -\frac{x}{\sigma^2 \sqrt{\pi}}\exp({-\frac{(x - \mu)^2}{2\sigma^2}}) $$ $$f'(0) = 0 \iff 0 =0$$ yep, no valuable ...
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How to get Variance from Gaussian distribution and Random Initialization

I am following deeplearning.ai's videos on Coursera. Prof Ng mentions that specific random initialisations for the weights(for example, by Xavier or He initialisations) can help optimise learning. ...
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1answer
26 views

Time Series assumptions for iid $\epsilon$

thanks for reading my post. I know its fundamental and rather easy qns but I'm seriously struggling. Please help me, thank you very much! Let $\boldsymbol{X}$ have a distribution with mean $\mu$ and ...
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15 views

Time Series Analysis: Determine Which Value Will Be Hit First?

I am analyzing a time series of sales data, and calculating probabilities of certain targets being reached with the assumption that the distribution is normal (Essentially use a ...
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2answers
257 views

Hypothesis testing- with normal approximation

In Europe the diameters of women's rings have mean 18.5 mm .Researchers claim that women in Jakarta have smaller fingers than women in Europe .The researchers took a random sample of 20 women in ...
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What is the intuition behind pi in the PDF of a Normal Distribution ? Is it related to some sort to a circle / sphere

The PDF of a Normal distribution is given as below I am aware of the various properties of Normal distribution and how the two parameters mu and sigma affect the shape of the distribution. What is ...
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Expected value for a function with a normal distributed random variable

I have a random variable $X \sim N(\mu, \sigma^2)$ and a function $5x^2 + 2x$. How can I calculate $E(g(x))$ ? I have two ideas, altough I'm not sure which one is right: $E(g(x) = \int_{-\infty}^{\...
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1answer
32 views

Expected Value of Random Vector [on hold]

I've got this exercise but cannot really understand the "logic" behind it. I have a random vector $t$ where $t \thicksim N(0, \sigma^2 I_n)$ and I've been asked to give: $E[t_i]$, $E[t_i^2]$ and $E[...
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Distribution of sum of cubes [duplicate]

If a set of $k$ random variables $x$ is drawn from the same normal distribution with mean $μ$ and standard deviation $σ$, then: the mean of the distribution of $Σx$ is $kμ$ the mean of the ...
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1answer
39 views

MLE for bivariate normal data with known error variances

I am working with a set of bivariate data arranged into columns labelled 'x' and 'y'. I also have measurements for the error variances corresponding to each observation, labelled 'sx' and 'sy'. An ...
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Why do we use $S^2$ while estimating the variance?

Sorry the title is a bit silly, but I currently confront a problem related to Fisher's information. Let $X_1, X_2, \cdots, X_n$ be of $N(\mu , \sigma^2 )$ distribution where $\mu$ is known, $U^2 := ...
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45 views

How to compute $E[ (|X|) X]$ when $X \sim N(0,1)$?

Any help would be appreciated.
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Sum of Two Bivariate Normal Distributions

I'm just confused on how to set up and start this problem. I'm confident that once I start down the right path, I'll have little issue. Let $p_1$ denote a bivariate normal distribution $N(0, 0, 1, 1, ...
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Which test is suitable in below situation?

I want to calculate the correlation of two continuous variables, unfortunately both of them are not normally distributed. (by using Shapiro-Wilk test) There are two categorical confounding variables ...
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Mean of repeated samples

Let's assume we have random variables $X_1\sim N(\mu_1,1),\cdots,X_n\sim N(\mu_n,1)$. Now we take one sample from each and get $X_1 = x_1,\cdots,X_n = x_n$. We order them and calculate the mean of top ...
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Sensitivity analysis of an equation in R language [closed]

I have the equation, Y = A*B*C/(D*E) Where A, B, C, D and E are the certain parameters of 1000 samples (say groundwater samples). ...
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1answer
51 views

linear combination and univariate normal

Show that $(X_1,X_2)$ has a bivariate normal distribution with means $\mu_1, \mu_2$, variances $\sigma _1^2 $ and $\sigma _2^2$, and correlation coefficient $\rho $ if and only if every linear ...
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1answer
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Bivariate normal distribution from independent random variables

Let $X_1$ and $X_2$ random variables such that $X_1+X_2$ and $X_1-X_2$ have independent standard normal distributions. Show that $x=(X_1, X_2)$ has a bivariate normal distribution. My work: Since $...
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Joint distribution of sample means from multivariate normal

Suppose $\mathbf{Z_1}, \dots, \mathbf{Z}_n$ be iid random vectors of normal distribution s.t. $\mathbf{Z}_i \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma})$ where $\mathbf{Z}_i = (X_i, Y_i)^T$, $\...
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Conjuage priors for linear combination

My knwoledge of statistics in general and Bayesian statistics in particular is limited. With that in mind, I would sincerely appreciate if somebody could help me with the following problem that I have ...
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27 views

What distribution can be predicted more accurately?

I am working on simple statistical prediction. However, whatever i did, i come up with a high range. I have used standard deviation and also percentile. Yes they work well in the given data but the ...
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1answer
37 views

How to find probability of quantiles given mean and variance? [closed]

If someone could explain the process for beginning this and the formulas involved I would be grateful. Male height in the Netherlands is normally distributed with a mean of 73 inches and a ...
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1answer
29 views

How normal is the following distribution of data? [closed]

I'm using the following dataset with 2 columns (features) and 1 label to train a Gaussian Naive Bayes classifier. How would you determine (using a stastiscal normality test) whether the data is ...
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2answers
47 views

Will changing the standard deviation affect the distribution?

Let's say I want to generate some random data that follows the normal distribution, with a mean of 5. Will setting the standard deviation to 3 or 5 affect the distribution, that is, will it still ...
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What is the standard deviation and mean of the reciprocal of normal distribution in terms of that of the normal distribution? [duplicate]

What is the standard deviation and mean of the reciprocal of normal distribution in terms of the standard deviation and mean of the normal distribution?
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Is it possible to get equivalences of Western Electric Rules in non normal distribution with percentile values?

I am not a statistician and I am a bit lost with these concepts, but I will try to explain my situation. I have a non-normally distributed data, and I am trying to apply the Western Electric Rules to ...
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Why does the marginal likelihood integral have no closed-form solution?

In Bayesian inference we end up with the formula: $$ P(\mathbf{w|t,X)}= \frac{P(\mathbf{t|w,X)}P(\mathbf{w)}}{\int P(\mathbf{t|w,X}) P(\mathbf{w}) d\mathbf{w}}$$ Assume the prior $P(w)$ is a ...
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Can I directly test Beta=0 with t-test if General Linear Model assumptions are met?

Dear Stackexchange users, I have a question regarding the use of t statistics on the beta values of a GLM. If I understand correctly, when computing a GLM, one has to check that: the residuals are ...
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1answer
19 views

Under what conditions should I use an approximate Z-score vs a t-test? [duplicate]

I am struggling to understand the limiting assumptions of simple hypothesis testing using Z and T statistics under different scenarios. In a case where X is normally distributed, and n > 30, and $\...
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2answers
51 views

How do I calculate the standard deviation of a normal distribution given the mean and a quantile value of that distribution?

How can I determine the standard deviation of a normal distribution with known mean and a known percentile value? The known percentile value would be on the correct side of the mean (eg, for ...
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1answer
29 views

Why Normal + Normal x Rademacher is not Normal?

I am coming back to a well-known example: let $X$ follow $N(0,1)$ and $T$ follow a Rademacher distribution $(p(T=1)=p(T=-1)=1/2)$. Then, it can be easily demonstrated that $TX$ follows also $N(0,1)$. ...
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Why is the normal distribution called “normal”?

It occurred to me that there is no question on here about the name "normal distribution" yet. There is this question, about whether to call it normal or Gaussian, but it does not address why it is ...
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1answer
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Why aren't all Normal and Rademacher variables independent?

In Wikipedia page: https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent, we find a classical counter-example showing that two normally distributed and ...
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1answer
25 views

How do I find the standard deviation that results in a specific probability coverage in a truncated normal distribution?

Given a truncated normal distribution $X$ with mean $\mu$, lower limit $a$, and upper limit $b$. How can I pick a standard deviation $\sigma$ such that $P(\mu -x\leq X \leq \mu+x)=y$ for some ...
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1answer
28 views

Box-Cox vs Yeo-Johnson

Both Box-Cox and Yeo-Johnson transform non-normal distribution into a normal distribution. However, Box-Cox requires all samples to be positive, while Yeo-Johnson has no restrictions. To me, it ...
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1answer
31 views

Literature on Bayesian stuff with Normal Distribution? [on hold]

I am writing something on Bayesian Analysis involving the normal distribution. I know that the conjugate prior is the so-called normalized Gamma inverse distribution, I know the update rule for the ...
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25 views

Difference between a Bayes classifier with diagonal multivariate gaussian class conditionals and a Naive Bayes classifier?

In a Bayes classifier, let's say we want to fit a multivariate Gaussian distribution for the class-conditional probabilities and we restrict its covariance matrix to be a diagonal matrix. In a Naive ...
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1answer
324 views
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1answer
17 views

Mahalanobis distance - understanding the formula [duplicate]

I've read quite a few explanations on this topic, liking this one the most: https://mccormickml.com/2014/07/22/mahalanobis-distance/ But there is still one thing I don't understand. I understand ...
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1answer
69 views

Suppose $X$ follows a normal mixture. In which cases is $X$ itself normally distributed?

Suppose $X$ follows a normal mixture with cdf: $F(x)=0.5\Phi(\frac{x-\mu_1}{\sigma_1})+0.5\Phi(\frac{x-\mu_2}{\sigma_2})$, where $\Phi(\cdot)$ denotes the normal cdf. Without further information, if ...
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0answers
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Distribution of the squared norm of a vector with multivariate normal distribution and dependent components [duplicate]

Let x be a p-dimensional random vector with dependent components. Assume that x is distributed according to a multivariate normal distribution with mean vector m and variance/covariance matrix V which ...
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1answer
19 views

Gibb's sampling where target prob distribution is itself a conditional joint distribution - p(x,y|t)

I'm new to Gibb's sampling and need basic guidance. Say p1,p2,q-> are Gaussian variables. p1->q<-p2 and q->x where x is a discrete variable (either 1 or 0). How do I go about Sampling (using ...
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Why is the limit of a Chi squared distribution a normal distribution?

My professor claimed that $\lim_{p\to\infty}\chi^2_p$ has a normal distribution. The claim was made on the basis of the Central Limit Theorem: as $p\to\infty$, we have a Normal$(p\mu, p^2\sigma^2)$. I ...
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1answer
63 views

Finding joint distribution of $(X + Y,X^2 + Y^2)$ where $ X,Y$ are independent standard normal variables

Find joint distribution of $W = X + Y$ and $Z = X^2 + Y^2$ where $X,Y \stackrel{\text{i.i.d}}\sim\mathcal{N}(0,1)$. I am trying to do this by the change of variable method. So first I need to get $X,...