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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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Distribution of the product of $n$ i.i.d log-normal random variables [duplicate]

Let $X_1,\ldots,X_n$ be independent log-normal random variables such that $$log(X_i)\sim N(\mu,\sigma^2)\ \ \forall i=1,\ldots,n$$ Then what can be said about the distribution of the random variable $...
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classify samples from two different gaussian distributions [duplicate]

Assume we have two sampling process, i.e. we can draw samples from two different gaussian distributions P and Q. At first, we draw samples from gaussian distribution P only. But after some time t, we ...
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Defining a covariance matrix [on hold]

I'm working with a Gaussian Process, and my covariance matrix is isotropic, i.e. it is defined just by the distance of the points locations. Suppose I'm working with Squared Exponential Covariance. ...
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2answers
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Covariance of two normally distributed variables

I saw in a statistic book that "It can be prooved that if two normally distributed variables have covariance = 0, they are independent". How can I start this proof? Can I say that $cov(X,Y) = E(XY) ...
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How to get a random function with a truncated normal distribution? [duplicate]

My goal is to have a function that given an input value will output a random value that follow a truncated normal distribution, could you please suggest a function that can do that ? Or guide me a ...
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1answer
23 views

Joint distribution of the magnitude/phase of a circular bivariate normal distribution?

A bivariate normal distribution with no correlation and identical variance in both dimensions can be written as $$ P(x,y|\mu_x, \mu_y, \sigma) = \frac{1}{2\pi\sigma^2}\exp{\left(-\frac{1}{2}\left(\...
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Using gaussian as prior probability distribution

I have seen in several places that people use a Gaussian distribution as the prior probability distribution. But when I am assigning probabilities, does using Gaussian a good idea, given that we can ...
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1answer
26 views

Goodness of fit test for a normal distribuition

I have the following exercise that shows $n=6$ numbers: $$ 1.40, 1.55, 1.35, 1.50, 1.29, 1.64 $$ Is data normally distributed at the 5% significance level? Surely $\overline{x} = 1.455$, $s=0....
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Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$

Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
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1answer
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A question about the Least Squares Estimation: what motivates its definition in the general case?

Let $Y_{1},Y_{2},\ldots,Y_{n}$ be independent random variables with expected values $\mu_{1},\mu_{2},\ldots,\mu_{n}$, respectively. Suppose that the $\mu_{i}$'s are functions of the parameter vector ...
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1answer
68 views

What does it mean exactly to divide a distribution by another distribution?

In the notes I'm working through, distributions are often "divided" by other distribution, and while I sort of understand what is meant, i would rather a rigorous explanation. Let me provide an ...
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Intuition behind sub-gaussian and sub-exponential random variables [on hold]

I'm having a hard time in grasping the meaning of sub-gaussian and sub-exponential random variables due to multiple equivalent definitions available on the internet. Can someone please give some ...
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1answer
23 views

Scaling different data sets such that transformed values will be between (0,1)? [on hold]

I am creating a model to calculate the weighted average score where parameter values are coming from different datasets. DATASET 1 RANGE: [-1,1] DATASET 2 RANGE: [0,100] DATASET 3 RANGE: (-...
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Concise overview of prototypical distributions

[This question is mainly a reference request.] I'm searching for a somehow concise and complete table of prototypical distributions that would allow a test person to easily choose which typical ...
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Probability of getting two $z$-scores that are within one standard deviation of each other

Randomly choose two $z$-scores following a normal distribution with mean 0 and standard deviation 1.* What is the probability that the $z$-scores are within one standard deviation of each other?
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1answer
30 views

Question about the Multiple Linear Regression: why and how does it work?

I know this question is quite simple and maybe quite naive as well, but I would like to get some help. The general linear model can be expressed as \begin{align*} \textbf{Y} = \textbf{X}\beta + \...
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Using information about optimising function in bayesian optimisation

I know that bayesian optimisation is a strategy for optimising black-box functions. But if i have some information about type of function or it's specific behaviour at some interval what methods of ...
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Is the Coefficient of Variation valid for data which does not follow a normal distribution?

I am trying to compare the dispersion of several data vectors. As an example I have that via two methods produces one vector of data that fits a normal distribution and other one that follows an ...
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1answer
99 views

Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$

How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
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2answers
110 views

Calculate the probability of mu for normal distribution [on hold]

Maybe this is a stupid question, but given we have drawn $n$ samples $x_1,...x_N$ from a normal distribution $\mathcal{N}(\mu, \sigma)$. How do I calculate the probability that eg $\mu=1$? ie $P(\mu=1|...
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2answers
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Identify a switch between probability distributions

Say I have two normal distributions with means $\mu_1$ and $\mu_2$ and standard deviations $\sigma_1$ and $\sigma_2$, respectively. A t-test reveals that the means are significantly different, but the ...
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How do we prove the bivariate normal distribution belongs to the exponential family?

Let $\textbf{X} = ((X_{1},Y_{1}),(X_{2},Y_{2}),\ldots,(X_{n},Y_{n}))$ be a sample from a bivariate normal population. Show that the distributions of $\textbf{X}$ form a five-parameter exponential ...
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2answers
42 views

R and mean value of random variable

This question might be a bit naive. According to theory the mean value of a r.v. is the sum of the value times the pdf. I try to test this in R. I am using the following code: ...
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1answer
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Meaning of Bullets within Normal Distribution

I am working through the derivation for a method published in this paper. On page 586 (equation A1) the authors use the notation: $$ N(\bullet,\bullet) $$ What do the bullets represent? Do they ...
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Are there any restrictions that we can use to ensure that a Gaussian distribution is ``close" to a given mixture of Gaussians?

Are there any restrictions that we can use to ensure that a Gaussian distribution is ``close" to a given mixture of Gaussians? Say you have $d + 1$ gaussian distribution with parameters $\theta_i = (\...
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Consecutive normal distributions influencing each other [closed]

I'm trying to model the probability of a disc brake needing replacement. All I know is that they wear out on average every 100.000km. I've figured that they need replacement according to a normal ...
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Likert Scale data analysis [closed]

I have a dataset where judges give singers scores on Likert Scale on various attributes. Now we are to find what their ranks should be. Now usually normal distribution is assumed, but my teacher asked ...
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Probability distribution question

An auto parts company, produces cylinder liners for engines of 1.2 inches in average diameter with a standard deviation of 0.1 inches. Every piece has a diameter less than an inch or more than 1.4 ...
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Expectation of normal RV conditional on normal mixture

Let $v\sim\text{Normal}\left(\mu,\sigma_v^2\right)$ a random variable with $\mu>0$ and $u\sim\text{Normal}\left(0,\sigma_u^2\right)$ Let $k\sim\text{Binomial}\left(N,p\right)$ a random variable ...
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How do I find the probability of interference of a shaft and a bearing given their nominal diameter values and their standard deviations?

The exact question is as follows: An assembling of shaft and bearing is made out of shaft manufactured to a specification of 30.00 ± 0.09 mm and bearings are manufactured to a specification of 30.10 ±...
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1answer
21 views

Usage of sample covariance and sample mean

I understand the difference between sample mean/covariance and population mean/covariance and how to calculate them. However, I'm a bit unsure about what happens afterwards. If I only have the sample ...
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Compare variables with zeta-score

Assume the calculation of a physical quantity $x$ from a given formula i.e. $$x=\dfrac{a*b}{c}$$ where $a, b$ and $c$ are experimental observables, therefore $x$ is a quantity derived from ...
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1answer
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When is the log-normal distribution appropriate?

Iv'e read the Wikipedia entry about the log-normal distribution, as well as a few other sources online, and still do not understand what sort of natural processes are expected to produce a log-normal ...
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Conditional probability question for healthcare scenario

I'm trying to determine the probability of an event occurring, given that another event has already occurred. Here's the scenario: Surgeons are allocated a block of time to perform cases. For ...
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Mysteries of the Normal Distribution [duplicate]

I have been studying ML for over a year and am actually a Bachelors of Statistics myself and am sick of not knowing the beauty of the Gaussian distribution and why it is so prevalent in nature. I've ...
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4answers
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Assuming a normal distribution: what is the sd for a given mean?

Assume that accordning to a national statistic people in germany have a $mean$ of $10,000$ € on their bank account and we call that their "assets". Unfortunattely, only the $mean$ is given but not the ...
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Poisson distribution for very large numbers - can i decrease the interval?

I want to set up a alert notification if transactions on our website fall above or below a low probability number as it could be due to an error. Lets say i have mean transactions per day of 50,000; ...
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1answer
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Normal distribution and the probabilities of type I errors

New pupils entering a large secondary school take a general knowledge test during their first week. The mean score achieved on this test is 46.7 with a standard deviation of 14.3. At the beginning of ...
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non homogonous data and interaction

I have an experiment design with: Variable 1 - 2 levels Variable 2 - 3 levels And demographic information collected about my participants: Demographic 1 - 2 levels Demographic 2 - 3 levels ...
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1answer
22 views

Integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$

I'm trying to integrate $\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{(-\frac{1}{2}(\frac{x^2}{4}+4y^2))} dy$ using the fact that the integral of any normal PDF is 1. But I'm having trouble completing the ...
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1answer
23 views

Conditiional probability as mean of gaussian?

I am reading the paper "Tutorial of Variational Auto-Encoder" and I faced the following notation: $$P(z) = \mathcal{N}(z\mid 0,I)$$ $z$ is a latent variable which should have been normal distributed. ...
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1answer
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Explanation of the Posterior Derivation of the Gaussian Distribution

I'm reading through my notes and I don't quite understand this bit: I understand how the likelihood was calculated but no more than that.Can anyone explain the steps and exactly how they go from one ...
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0answers
32 views

Understanding KL divergence between two univariate Gaussian distributions

I'm trying to understand KL divergence from this post on SE. I am following @ocram's answer, I understand the following : $\int \left[\log( p(x)) - log( q(x)) \right] p(x) dx$ $=\int \left[ -\frac{1}...
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How to calculate the percentage of values that lie within some range of Standard Deviation around the Mean in a Normal Distribution Curve) [duplicate]

This wiki article on 68–95–99.7 rule states - In a Bell Curve (i.e Normal Distribution Curve), 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, ...
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Approximating the Kullback-Leibler Divergence with a Laplace approximation

Suppose I wish to compute the (asymptotic) Kullback-Leibler Divergence (KLD) between the exact Bayesian posterior $$q_{n}(\theta|x_{1:n}) \propto \pi(\theta)\prod_{i=1}^n p(x_i|\theta)$$ and the ...
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1answer
36 views

Consistency of the maximum likelihood estimator for the variance of a normal random variable when the parameter is perturbed with white noise

Let $X_1, X_2, \dots , X_n$ be normally distributed independent observations with known variance $\sigma^2$ and mean respectively given by $\mu_i = \mu + \epsilon_i$ where $\epsilon_i$ is white noise, ...
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1answer
21 views

Is it correct to do stats in log transformed metabolomics data?

I have a dataset from targeted metabolomics analysis, the units I am working with are ng/ml[creatinine] (I use creatinine concentration to normalize the data since the samples are urine and can have ...
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1answer
39 views

Estimate parameters of a normal knowing the density function

I have a matrix with 3 columns. The first column contains values of a variable $x_1 \in [-1,1]$. The second column contains values of a variable $x_2 \in [-1,1]$. The third column contains a variable $...
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1answer
33 views

Question related to Wald and Wolfowitz paper on tolerance limits

I have a question regarding Wald and Wolfowitz paper on tolerance limits for a normal distribution. There paper can be found here Wald and Wolfowitz denote the root of the equation: $ A(\bar{x},s,\...
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3answers
108 views

Why is there a change in the number of degrees of freedom when the following modification is made?

In the notes that I'm working through it says the following: "Let $X_1,...,X_n$ be a random sample from $N(\mu,\sigma)$ $$\sum^{n}_{i=1}\Bigg[\frac{(X_i-\mu)}{\sigma}\Bigg]^2$$ has a $\chi^2$ ...