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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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Can I split a normal distribution in several smaller ones? If so can I just split the SD? [on hold]

Let's say I got a time that follows a normal Distribution with a mean of 30 minutes and a sd of 6 minutes. Now I want to cut this distribution into three parts that all have a mean of 10 minutes. If ...
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1answer
41 views

Confidence intervals over mean difference with unknown but equal variance

The problem is the following: I have 2 Stores, called Store1 and Store2 I take a sample of the number of item sold in each store over a certain period of time From the first store the sample has ...
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25 views

Simulation: Generate random numbers that cluster around an average? [on hold]

I want to simulate a simple event that has variable empirical result/outcome. For example, let's say we collect the data for how far people can throw a ball. The data may or may not be distributed ...
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14 views

Calculating gaussian integrals with sympy [on hold]

I want to calculate gaussians integrals like: $$\int \mathcal{N}(x_0 \mid \mu_0 + t \mu_1, \sigma_a) \mathcal{N}(t \mid \mu_b, \sigma_b) \mathrm{d}t$$ where $\mathcal{N}(t \mid \mu, \sigma) = \frac{...
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24 views

Is Wikipedia wrong about the Multinormal Distribution PDF

I was on the Wikipedia page of Multivariate Normale Distribution and I was wondering if the PDF of the Multivariate Normale Distribution is right. Shouldn't be: $(2\pi)^{(-k/2)}|\Sigma|^{-(1/2)}$ ...
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29 views

Determining if two time series are cointegrated

I'm currently working on a problem where I am given that $\epsilon_t$ is a series of independent draws from a N(0,1) distribution $w_t$ is another series of independent draws from a N(0,1) ...
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1answer
29 views

Analytical results for sum of normally distributed random variables with e.g. sorting?

Assume I have bolts and holes, both following a normal distribution with $$\mu_B = 90 mm, \sigma_B = 8 mm$$ $$\mu_L = 100 mm, \sigma_L = 4 mm$$ As we know that the sum also follows a normal ...
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1answer
74 views

Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
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30 views

What is the correct formula for Bayesian update for normal distribution with known variance [duplicate]

As question title states, I'm interesting in Bayesian update of normally distributed data with known variance. I compared three sources and they seems to contradict each other. I use some kind of ...
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1answer
40 views

Generative autoencoders - how important is agreement of latent variable distribution e.g. with Gaussian?

Autoencoders want to minimize distortion of encoding-decoding process, preferably alongside evaluation by discriminant. Generative autoencoders additionally would like latent variable from a chosen ...
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31 views

How to infer the mean and sd of a gaussian given two quantiles [on hold]

I have a normal distribution and it is given that P(x<1.5)=0.24 and P(x<2.5)=0.95. How do I calculate mean and standard deviation? A code example in R would be welcome?
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1answer
29 views

How do I compute the closed form normalizing constant for this distribution?

The funnel distribution for random variable $X = (x_1,x_2,..,x_D)$is $$P(X) = N(x_1|0,9)\prod_{d=2}^D N(x_d | 0,exp(x_1))$$ The closed form normalizing constant for normal distribution $N(x|0,\...
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A Normal random variable divided by a Chi-squared random variable

I am looking to find the pdf of the ratio $Z = \frac{X}{Y}$, where $X \sim$ Gaussian and $Y \sim$ Chi-squared are indepedent random variables. A reference is good enough if you do not want to write ...
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34 views

Estimating sample variance

Suppose that we have 2 independent samples $X_{11}, X_{12},.., X_{1n_1}$ and $X_{21}, X_{22},.., X_{2n_2}$ from a normally distributed population with $n_1<n_2$. Does that mean that the sample ...
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2answers
46 views

How to compute the variances of quotients of normal variables?

$x$ and $y$ are independent normal random variables. $z1=x/(x+y)$ and $z2=y/(x+y).$ How to obtain the variances for $z1$ and $z2$? I understand through the delta method, $\operatorname{Var}(z1)$ ...
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Finding sample sizes [closed]

nine out of 25 randomly selected students of WMU live in Southwest Michigan. The result says that the true proportion could be as low as 0.17 or as high as 0.55. If we want to reduce the margin of ...
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1answer
40 views

Easiest proof of the well known normal result

The well known identity that for iid normal distribution with $x_{i} \sim N(\mu, \sigma^{2})$ we can write $\sum_{i=1}^{n} (x_{i}-\mu)^{2} = s^{2}+n(x-\bar x)^{2}$ where $s^{2}=\sum_{i=1}^{n}(x_{...
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0answers
12 views

Sum of the normal random variable [closed]

I suppose a set of independent normal random variables ${x_i} \sim {\cal N}({\mu _i},{\sigma _i})$, $Y = \sum {{x_i}}$. How to get the function of Y(z) with the conditions of ${x_i} > 0$,$0<\sum ...
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2answers
52 views

Can a Bernoulli distribution be approximated by a Normal distribution?

$$\sum_{i=1}^n bernoulli(p) = binomial(n,p) \approx \mathcal N(np, np(1-p)) = \sum_{i=1}^n \mathcal N(p, p(1-p))$$ Can I conclude that $\mathcal N(p, p(1-p))$ could represent an approximation of $...
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1answer
30 views

Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
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0answers
9 views

Visual basic code [closed]

I want to produce 88 random numbers giving the sum equal to 440. With the following constraints. The numbers are between 0 to 20 (without 1) The numbers are integers I am searching for a VBA code ...
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1answer
22 views

Concise way to visualize / compare many Gaussian mixtures

I have 5,000 samples drawn from each of approximately 50,000 distributions. I have good reason to expect most of them to be normally distributed, and I expect some of them to be multi-modal (mixture ...
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1answer
34 views

Multivariate t approach multivariate normal

It is well known that in univariate case, as we increase the degree of freedom in a t distribution, it will limit to normal distribution. Does the result hold true for the multivariate case as well?
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2answers
39 views

How to swap variables in a conditional normal distribution?

I assume that I have two normal distributed variables where one depends on the other: $P(A) \sim N(0,\sigma_a)$ $P(B|A) \sim N(q\cdot A, \sigma_b)$ How can I get the reverse $P(A|B)$ assuming that ...
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0answers
13 views

Calculate the Confidence Interval for the Error of Model

I am not sure I am thinking about my problem the right way, so I am looking for the right approach. I have a data set that, for the sake of argument, has a mean of 1 and a standard deviation of $\...
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1answer
69 views

Confidence interval for $\sigma^2$

I started with any distribution and underwent the CLT on $\sqrt{n}(\widehat{\sigma}^2 - \sigma^2)$ where $$ \widehat{\sigma}^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \mu)^2 $$ is a sample mean of $\sigma^2$...
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1answer
50 views

Understanding the shifted log-normal distribution

I have difficulties understanding why a third parameter (the shift) is necessary to describe the log-normal distribution. Let's say we have a normal random variable X, if I shift this variable by an ...
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1answer
43 views

Do we assume a t distribution for the estimate of the difference of normal distributions?

I am studying hypothesis testing. When performing a one sample t test, we assume a t distribution for the sample mean estimates of the true mean. When conducting a two independent sample test, it ...
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What does the covariance / inverse Fisher do in Large sample theory?

What does the covariance / inverse Fisher do in Large sample theory? $$\hat{\theta} \sim N_p(\theta, I^{-1}(\theta))$$ It's in the place of variance of normal distr. However, the $I^{-1}$ gives $Cov(...
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2answers
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Generate Data where outcome is conditional on independent variables [closed]

I want to generate a synthetic dataset {Y, X1, X2}. Independent random variables X1 and X2 follow bernouli distribution where probabilities for X1 and X2 are known. Whereas, outcome variable Y needs ...
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1answer
17 views

Confidence interval for the square of binomial probability

I have a binomial distribution where the estimate for p is 0.03 out of 1000 sample trials. Using the normal approximation and Chi-square distribution for the square of normal distribution, how can I ...
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1answer
67 views

A question about Trivariate Normal Distribution

Question: Let $(X_1,X_2,X_3)\sim N_3\left[\mathbf0, \begin{pmatrix}1&\rho_{12}&\rho_{13}\\\rho_{12}&1&\rho_{23}\\\rho_{13}&\rho_{23}&1\end{pmatrix} \right]$ Show that ...
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1answer
36 views

Show that target variable is gaussian in simple linear regression

Given the simple linear regression model $$ y_i = \beta_0 + \beta_1 x_i + \epsilon_i$$ where $\beta_0$ and $\beta_1$ are fixed paramters, $x_i$ are nonrandom variables and the errors $\epsilon_i$ ...
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2answers
121 views

Deriving an expression for a confidence interval for σ^2 using the asymptotic distribution of √n(σ̂^2−σ^2)

We have We have $X1,…,Xn i.i.d N(μ,σ^2) $where $μ$ is known and $σ^2$ isn't known. $σ̂^2=(\frac{1}{n})∑(X_i−μ)^2$. First of all what I did, I derived an equitailed 95% confidence interval for $σ^2$....
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3answers
119 views

How to derive $\operatorname{var}[(X_i−\mu)^2]=2\sigma^4$ where $X$ is distributed normally

I have $X_1,...,X_n$, i.i.d. $N(\mu,\sigma^2)$ and I would like to calculate $\text{var}[(X_i−\mu)^2]$. I know that the solution is $2\sigma^4$. However, I can't derive it. Any suggestions?
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28 views

Determine overlap of mixture models

I have a modeling problem where the time between events can be approximated as a mixture model of 2 Gaussians ($\mu=14, \sigma=5, \lambda=0.8$ and $\mu=6, \sigma=2,\lambda=0.2$). The response to ...
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0answers
11 views

Sums of degenerate quadratic forms

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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1answer
39 views

Log transformation to generate random number producing NA's

I am trying to generate a random values using log distribution. The reason for using log-distribution is keep the values positive. ...
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1answer
135 views

Proving Asymptotic distribution of $\sqrt n( \widehat\sigma^2 -\sigma^2)$

I am looking at trying to derive an expression for the asymptotic distribution. We have $X_1,\ldots, X_n$ i.i.d $N(\mu, σ^2)$. So we have defined $\hat \sigma^2 = \frac 1n \sum_{i=1}^n(X_i-\mu)^2$. (...
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0answers
31 views

How to aggregate multiple Z-scores

Is there a way to aggregate multiple Z-scores to get a single Z-score that corresponds to the probability that no null hypotheses are rejected? Backstory: We have a tool that has some error E that ...
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2answers
195 views

Expectation of Inverse Logit of Normal Random Variable

I have a random variable $Y = \frac{e^{X}}{1 + e^{X}}$ and I know $X \sim N(\mu, \sigma^2)$. Is there a way to compute $\mathbb{E}(Y)$? I have tried to work out the integral, but haven't made much ...
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2answers
53 views

Sampling distributions seem to be quite useless

I am studying estimation and found the concept of sampling distribution hard to grasp. The book I am reading claims that "sampling distributions" answers the following question: how confident should ...
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0answers
23 views

Does Probability and P-value same thing? What is actually P-value means in hypothesis testing (t/ ANOVA / Tukey)? [duplicate]

We have been taught that P-values is "calculated probability". In this past 10 years, I have been using statistical test (t/ ANOVA / Tukey) more than thousand times. I have published scientific papers ...
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0answers
15 views

llr for log(sigma^2)

The llr for the variance of a normal distribution is very dissimilar to a quadratic, and hence (I think) we can say that the Wald Test will work poorly. I found that by instead investigating the ...
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0answers
41 views

Obtain a random sample from a sum of two dependent random variables

Suppose $X$ and $Y$ are dependent random variables and I know the marginal densities of $X$ and $Y$ (if simple we can assume they are Gaussian). Using a copula I may be able to estimate the joint ...
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1answer
23 views

Binomial and Normal distribution X~N(18,5^2) [closed]

I'm stuck with a problem related to binomial and Normal distribution. Suppose the amount of time a laptop battery holds its charge is: $X \sim N(18,5^2)$ Suppose there are 40 students and there is ...
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0answers
17 views

Are these scores normally distributed?

Is this Q-Q plot interprets the normality of the ARAT scores? What is the result?
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1answer
28 views

95% limits of a normally distributed parameter

How do I find the 95% limits of the population distribution of a normally distributed parameter? I've taken the mean and SD from 10 different readings of the parameter. Will the 95% limits be mean +/-...
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0answers
13 views

Uniform distribution on $l_1$ ball of $R^n$ is not Sub-Gaussian when n is a variable?

I am self study the high dimensional probability by Roman Vershynin I was asked to show that : Uniform distribution on $l_1$ ball of $R^n$ is not Sub-Gaussian, on page 55. Exercise 3.4.9 Here is ...
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0answers
16 views

Explain the results of the 2-sample Kolmogorov-Smirnov test for two random Gaussian sets of numbers

Suppose I have two sets of random numbers drawn from the Gaussian distribution, $\mathbf{r_1}$ and $\mathbf{r_2}$. I then do a 2-sample Kolmogorov-Smirnov test (KS test) between the two to determine ...