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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How to develop a model in R to illustrate that sum of two variables normally distributed if these two variables each follow a Normal distribution?

I was wondering how can I use code in R to set up a simulation to show that if two random variables each follow a normal distribution with given means and variances, their sum is also normally ...
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Distributions underlying purchasing decisions

I am trying to simulate a list of transactions of the type that would appear on the bank statement, eg ATM visits rent/food payments, salary deposit, etc. For generating random transactions the focus ...
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Distribution of $\frac{1}{X}$ if $X\sim N(0,1)$

Given a random variable $X$ that has a normal ditribution with mean $\mu$ and standard deviation $\sigma$, what is the distribution of $\frac{1}{X}$? I guess that it like a normal distribution ...
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How close a sample is to the Normal distribution ( Berry-Esseen Theorem)

My question is how can I use the Berry-Esseen Theorem to know how close to the Gaussian distribution is $L$, where $$L=nLn(2)+Ln(r_1)+Ln(r_2)...Ln(r_n).$$ $r_i \geq 0$ is a i.i.d. random variable ...
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Trouble adding a normal curve to histogram in R [migrated]

I'm trying to add a normal distribution curve to a histogram using curve. This is my code: hist(df$col, freq=T) curve( dnorm(x, mean=8.9,sd=5), 0, 30, add=T, col="blue") The histogram does look fine,...
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Solving for the discriminant function in LDA

This is related to the question posted in: The discriminant function in linear discriminant analysis In the one dimensional case, where $p_k(x) = \dfrac{f_k(x)\pi_k}{P(X = x)}$, where $f_k(x) = \...
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45 views

Sum of log distribution

Suppose $x_i\sim N(0,1)_\mathbb{C}$ is a i.i.d. random variable from a complex distribution and $$P=2^n|x_1| |x_2|...|x_n|$$ Rewriting $x_i=r_ie^{i\theta}$; that is, $$P=2^n r_1 r_2...r_n,~r\geq ...
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Why use the student's t-test rather than z-score?

Suppose we are given IID r.v's $X_1, \ldots, X_n$ that are not necessarily normally distributed. Mean $\mu$ and standard deviation $\sigma$ are unknown and we want to construct a confidence interval ...
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Getting random number from Weighted sum of Normal distribution functions [duplicate]

I've a weighted sum of the 2 Gaussian distribution functions as below. How can I get a random number based on this sum of functions. The number of functions can vary up to 10.
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What does $N(x|\mu, \sigma^2)$ mean?

I am supposed to show that $f(x) = \sum_{k=1}^{K}\pi_k N(x|\mu_k, \sigma_{k}^2)$ complies with the properties of a density function but I have no idea how to do this since I am not sure what $N(x|\...
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Empirical risk minimization (ERM) solution to linear gaussian model

I'm reading the following paper on Invariant Risk Minimization https://arxiv.org/pdf/1907.02893.pdf. The toy problem they consider in the paper is Linear Gaussian with X1 being the causal variable and ...
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Derivative of signal is normally distributed - why?

I stumbled upon something while analyzing some of my data and don't know the answer to this. There seems to be some sort of drift in my measurement system and I wanted to know more about it. I ...
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Proof of identifiability

I have a random variable $R$ that takes values in ${1,2,3,4}$, and the conditional distribution of $R$ given each $(x_{1},x_{2})$ is given by the following formulas ($P_{i}(x_{1},x_{2})$ is just the ...
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“Variance is independent of class” assumption

Reading from Logistic Regression slides by Erdem, before deriving the linear boundary equation from the GNB equation it says "What if we assume variance is independent of class i.e. $\sigma^2_{i,0}=\...
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How to interpret a Log Normal Distribution

I have a dataset with 3 columns that are found out to be log-normally distributed. I am a little bit confused about how can I draw the conclusion in a log-normal distribution similar to Normal ...
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“Scoring Data”: How to determine thresholds of a column value so that resulting groups (1-10) have a normal distribution of members within the groups [closed]

I have a list of customers and I am trying to assign a score (between 1 and 10) to each customer based on their spend. I wish to find spend-thresholds for each score so that the resulting scores have ...
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How to adjust/normalize/standardize mean? [closed]

I am making a reviews/ratings section for a website, with ratings that range from 0-5 stars. I am not confident that the users of this system will all have the same idea of what these stars mean, so I'...
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Limiting distribution of maximum of i.i.d. Gaussians with decreasing variance

Consider a random vector $X^{(m)} = (X^{(m)}_1,\dots,X^{(m)}_m)$ where, for fixed $m$, the elements of $X^{(m)}$ are i.i.d. $\mathcal{N}(0,\sigma^2 / m)$. Define $$Z_m =\max_{k=1,\dots,m}X^{(m)}_k.$$...
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Log Transforming My TS Data for a First Difference Regression

I'm currently working with a ts of monthly yields where $Yield = \frac{Expense}{Blance}$. I am trying to understand the change in yield given a change in the market rate. My regression is $Y = \...
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Are normally distributed sample means equivalent to normally distributed residuals?

According to t-test:Assumptions "The means of the two populations being compared should follow normal distribution" The one way ANOVA test in case of 2 groups equals the t-test but the ANOVA ...
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bivariate normal distribution meaning [duplicate]

Does bivariate normal distribution mean the two random variables have normal distributions? is that enough for two random variables to have a bivariate normal distribution or are there some other ...
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Variance of backward looking system, autocorrelation

Say, I have a system, where $e_{i1}=\epsilon_1$ and $i=1,2,3,...,N$ and $e_{it}=f(t)\bar{e}_{-i,t-1}+\epsilon_t$, where $\bar{e}_{-i,t-1}$ is the average of the previous non-$i$ realisations. Also, ...
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Appropriate to fit lognormal model to data with heavy tail?

I am attempting to standardize recreational fishery CPUE data. I am using a delta approach, with a binomial model fit to the presence/absence data and a lognormal model fit to the positive ...
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Multivariate normal distribution transformation

Suppose that $X $ has a multivariate normal distribution $X\sim MVN (\mu, \Sigma) $, How can I transform $X$ into $Z$ so that $Z\sim MVN(\mu, I) $ where $I$ is the identity matrix? For instance, ...
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Multiplying bivariate gaussians by a constant

Say I have the following : $$ (X, Y) \sim N_2(\mu, \Sigma) $$ Then what would be the distribution of $(2X,2Y)$ ? Let $\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\...
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Bivariate normal probability of being inside ellipse

Assume that $\mathbf{X}$ is a bivariate normal random variable $$\mathbf{\mu} = E\mathbf{X} = \begin{bmatrix} 0 \\ 2 \end{bmatrix} \ \text{and} \ \Sigma = Cov \ \mathbf{X} = \begin{bmatrix} 3 & 1 ...
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characterization of circularly symmetric complex Gaussian variable

Let $Z=X + jY$ be a complex random variable, where $X$ and $Y$ are real. By definition, $Z$ is said to be circularly symmetric complex Gaussian $\mathcal{CN}(0, \sigma^2)$ if $X$ and $Y$ are ...
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What a normal curve actually is?

I am fairly new to Stats and thus this question. I was going through different materials to learn stats. Somewhere it's mentioned that a population, if plotted against frequency on a bar graph can ...
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Limit of $t$-distribution as $n$ goes to infinity

I found in my intro to stats textbook that $t$-distribution approaches the standard normal as $n$ goes to infinity. The textbook gives the density for $t$-distribution as follows, $$f(t)=\frac{\Gamma\...
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Max value of a pdf

We have $f_x(x;μ, σ^2) $ as the pdf of a normally distributed variable $X$. What is the maximum value of the pdf? I thought because the pdf is normally distributed, so it must be the case that mean ...
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the variance of a gaussian PDF?

A problem is this: The probability density function of the univariate Gaussian with mean $ μ $ and variance $σ2, N(μ,σ2)$: $$f_x(x) = \frac{1}{\sqrt(2*pi*σ2)} * e^-(x-μ)^2/(2*σ2)$$ The pdf of a ...
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How to find a such that X + aY is independent of X-aY for a bivariate distribution [closed]

Suppose $X$ and $Y$ are bivariate normal with equal variance, i.e. $[X, Y] \sim \mathcal{N} (0, \Sigma)$, where $$\Sigma=\begin{bmatrix}1&\rho\\\rho&1\end{bmatrix}$$ Find $a ≥ 0$ such that $...
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How to calculate the minimum of the difference between the upper and lower bounds of an interval?

Let's say $X$ ~ N(500, $50^2$) and $P(a<X<b)=0.95$. How can I calculate minimum(b-a)?
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Probability that the same r.v. generates the rth order statistic in one noise-added set, and the sth order statistic in another noise-added set

(Note: The title is confusing, as I have no idea if a name / short description exists for the setting below. I'm open to pointers and/or suggestions.) Setting Let $X_1, ..., X_N \overset{i.i.d.}{\...
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How to make a Median absolute deviation of $N(0, \sigma^2)$ an unbiased estimator of $\sigma$, asymptotically?

I am looking for a derivation of the fact that $\frac{1}{\Phi^{-1}(3/4)}$ is the multiplier needed for the Median Absolute Deviation (MAD) to be an unbiased estimator of $\sigma$ when $x_i\sim N(0, \...
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Independent samples test for mean without normal distribution with T-score

I have two independent samples of male and female order value. Female number of observations = 26887 Male number of observations = 12928 Female mean order value = 133.03 Male mean order value = 145....
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Logit transformation of (asymptotic) normal random variable also (asymptotically) normally distributed?

That is, if X is a (asymptotic) normal random variable, is $ln\left(\frac{X}{1-X} \right)$ also (asymptotically) normally distributed? From this question, i suppose it isn't the case but this paper ...
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Normal distribution in nature: additive result of multiple variables? [duplicate]

We found normal distribution is so common in nature, such as many measurement of species (weight, height or size). From the point of central Limit Theorem, Can I intuitively understand this as the ...
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Expected value of the SHASH distribution?

The sinh-arcsinh (SHASH) distribution has a pdf as follows: $f(x) = {\delta cosh(\omega)\over \sqrt{1+({x-\theta \over \sigma})^2}}\phi[sinh(\omega)]$ where $\omega=\gamma+\delta sinh^{-1}({x-\theta \...
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Confidence Interval for Difference in Proportions - Mistake in Question?

Question #3 of University of Michigan's Data Science Statistics Sample Questions: Applying the formula is straightforward, but it seems that the note at the end of the question is wrong. p1 seems ...
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MLE of variance is biased in a Gaussian distribution

Referring to: How to understand that MLE of variance is biased in a Gaussian distribution at some point during calculation the formula of the sum of the expected value becomes a single expected value:...
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Normalise scoring

I'm trying to solve the following problem. Multiple people score multiple vendors. The problem is that some people are less critical than others which results in them giving higher scores in general. ...
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How can I show that $(\bar{X}, S^2)$ is independent of $(X_{(n)}-\bar{X})/S$?

Let $X_1,\ldots,X_n$ be a random sample from $N(\mu,\sigma^2)$ with both parameters unknown. How can I show that $(\bar{X}, S^2)$ is independent of $(X_{(n)}-\bar{X})/S$? Work: I am quite confident ...
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Hypothesis testing: $H_0: \mu_1 \le \mu_2 \text{ vs } H_1: \mu_1 > \mu_2$ from two random samples

Let $X_1,...,X_n ∼ N(\mu_1, \sigma_1^2 )$ and $Y_1,...,Y_m ∼ N(\mu_2, \sigma_2^2)$ independent random samples with unknown parameters. Suppose we want to test the hypothesis: $$H_0: \mu_1 \le \mu_2 \...
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Probability that one Normal RV is greater than at least one of a set of other Normal RVs

For a side project I am working on I need to find the probability that a given Normal rv from $N(\mu_a,\sigma_a^2)$ is greater than or equal to at least one of the values from a set of other, unique, ...
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The second order moments of the Gaussian multivariate distribution

I am really quite confused to how to calculate and get the result from to I know the symmetry will recduce the sum of j to sum of i(yiyj to yiyi),but however, it is still difficult for me to think ...
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Expected value of the exponential of a normal CDF

Let $x$ be distributed according to a standard normal. Let $\Phi(y)$ denote the cumulative distribution function of a standard normal evaluated at $y$. Let $a$ and $b$ be real scalars. It is easy to ...
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NaiveBayes: Adding a new predictor which is random noise

In the NaiveBayes method, can adding a random noise vector where each element is sampled from, e.g., a standard normal distribution, help? In what cases this may be a 'clever' approach? I imagine ...
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Second order moment of multivariate Gaussian-completeness of the set of eigenvectors (bishop p. 83)

I don't know how 2.60 euqation is derived. Even if I have read the contents in previous pages, it is still hard for me to get an idea of how the 2.60 euqation is calculated. Also, I don't know the ...
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58 views

Cumulative distribution function of the posterior predictive distribution of a Gaussian process

Say I have calculated a posterior predictive distribution using Gaussian process regression. How can I assess the probability that an input X is less than or equal to some arbitrary value x? Is the ...

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