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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Joint probability of dependent random variables

I am not sure the following derivation is correct Let $Y \sim N(\mu_y, \sigma_y^2)$, $\theta \sim N(\mu_\theta, \sigma_\theta^2)$, and $x=y\theta$. Since $X$ and $Y$ are dependent, the joint pdf is $...
1 vote
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forming confidence ellipse versus using contrast

Suppose I am testing a hypothesis on data that is bivariate normal, $(x,y)\sim N(\mu,\Sigma)$ where $\Sigma$ is known but $\mu$ is not. My null hypothesis is a contrast $a^T\mu=a_1\mu_1+a_2\mu_2=0$. ...
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How to bound the probability of multivariate gaussian vector norm?

Let's say $v \in \mathbb{R}^n \sim \mathcal{N}(0, \sigma I)$. That is, $v$ is a gaussian random vector, whose entries are distributed $\mathcal{N}(0, \sigma)$ i.i.d. From the book "C. Giraud. ...
1 vote
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+50

Mutual Information between the mean of normally-distributed random variables and another variable

Given normally-distributed (possibly dependent) random variables $X_1 \dots X_n$, their mean $X=\frac{1}{n} \sum_iX_i$, and another discrete r.v. $Y$, can we relate $MI(X,Y)$ with the individual $MI(...
0 votes
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Calculation of conditional variance for 2 correlated random variables

If we have a bivariate normal distribution for $\left(X, Y\right)$ then, we can calculate the conditional mean and variance of $\left( X|Y \right)$, as demonstrated here https://online.stat.psu.edu/...
2 votes
2 answers
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Confused about Probability Density Function and Cumulative Density Function?

When computing probabilities, do we use probability density function or cumulative density function for continuous values? And I heard that if we have a cumulative density function for a set of ...
12 votes
3 answers
1k views

If a sample is normally distributed, is its population always normally distributed?

I know this is correct: "If a population is normally distributed, a sample randomly chosen from it must be normally distributed." For instance, If the sizes of all apples of my farm are ...
2 votes
1 answer
59 views

Is my data normally distributed? (QQ plot and histogram analysis) [duplicate]

I am trying to create a regression model for prediction. I need to generate prediction/confidence intervals for my model. I am trying to decide whether to use a quantile regression or linear ...
3 votes
0 answers
21 views

Equality in Gaussian Poincare Inequality

The Gaussian Poincare inequality states that: for $f: \mathbb{R}^n \to \mathbb{R}$ and $Z\sim \mathcal{N}(0,I)$, we have that \begin{align} Var(f(Z)) \le E[ \| \nabla f(Z)\|^2]. \end{align} My ...
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1 answer
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What are the main difference between a QQ plot and a probability plot for measuring nomality? [duplicate]

I am trying to evaluate the normality of the distribution of my model's residuals. I have been using statsmodels.api.qqplot and ...
4 votes
1 answer
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Unbiased estimator for $\mu_1/\mu_2$

Let $X_1,X_2,\ldots,X_n$ and $Y_1,Y_2,\ldots,Y_n$ be independent random samples from $N(\mu_1,1)$ and $N(\mu_2,1)$ populations respectively with $\mu_2\neq0$. I need to find an unbiased estimator for $...
16 votes
1 answer
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Generating samples from singular Gaussian distribution

Let random vector $x = (x_1,...,x_n)$ follow multivariate normal distribution with mean $m$ and covariance matrix $S$. If $S$ is symmetric and positive definite (which is the usual case) then one can ...
1 vote
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Singular distribution via push forward

Suppose $X$ is a multivariate normal distribution in $\mathbb{R}^3$ with a covariance matrix whose rank is 2. Therefore, it is a singular distribution. Is it possible to represent it as a push forward ...
2 votes
1 answer
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DFA in SPSS: Sorts effectively, but Box's M is still 0.000. Is the analysis worthless?

I am a geologist attempting to apply the discriminant function analysis to surface features I have mapped in ArcGIS. At the moment I have 4 dimensionless sorting variables calculated for each feature, ...
3 votes
1 answer
442 views

Numerical computation of the means and covariance in a truncated bivariate normal distribution

How can I compute the means and covariance of a truncated bivariate normal distribution? I am particularly worried about the case when the truncation occurs very far from the mean. Is there a robust ...
4 votes
0 answers
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Unbiasing estimator of $\|\Sigma\|_F^2$

I have access to samples of some distribution with second-moment matrix $\Sigma=E[xx^T]$ and need an estimate of $\|\Sigma\|_F^2$ (which can be used to set optimal size for LMS) We can use Frobenius ...
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Expected magnitude of cosine similarity among $B$ Gaussian vectors?

Take an IID sample of $B$ vectors $x$ drawn from a Gaussian with mean $\mu$ and covariance $\Sigma$. Define the following: $$S=\frac{1}{B^2}\left\|\sum_i^B x_i x_i^T\right\|^2_F$$ $S$ gives the ...
1 vote
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Does the number of samples, as opposed to the sample size in each sample, matter for the Central Limit Theorem? [closed]

(1)So here is a formula that describes CLT I found at https://en.wikipedia.org/wiki/Central_limit_theorem. According to the first part of the explanation, n as in Xn describes the number of samples(i....
2 votes
1 answer
22 views

GAM with opposite outcomes with different families

I'm building GAMs and I have some doubts regarding the family to use. I'm fitting GAMs because I expect some non-linear relationships between the response variable and some covariates. I've checked a ...
1 vote
1 answer
286 views

BIC in practice with gaussian distribution

I am considering a Gaussian distribution: \begin{equation} y \sim N(\text{net}(x,w), \sigma^2). \end{equation} where $\text{net}()$ is just the output of some neural net with weights $w$ and input $x$....
0 votes
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Help understanding lilliefors test

I need some help understanding the meaning of the lilliefors test. As far as I know, the lilliefors method provides a measure of normality of a data set. That is, a measure of how the images (values) ...
0 votes
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Which statistical Test should I use/ Comparing 2 groups

I am new to statistics, can someone guide me regarding the test I should use to compare the foot dimensions of 9 subjects to footwear dimensions (we have 2 footwear brands). So, to simplify, I have ...
1 vote
2 answers
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Sequential Bayesian updating of mean and variance of normal distribution

I am trying to write some code to learn the parameters of a normal distribution. I am new to this, and I have patched together the equations from various sources, which may be part of the problem. In ...
0 votes
0 answers
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Moving from Sample data to population

I have a data sampling set of success/failure data so I have classified it as discrete binomial data. ie. failure rate sampling 0/10 1/10 1/10 0/10 and therefore I can use the binomial distribution on ...
0 votes
0 answers
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Chi-square distribution, Chi-square distance

I am trying to use chi-square distance to find the similarity between two PDFs. My data is chi-square distributed. though the chi-square distance works well but my question is can I use the jensen-...
163 votes
9 answers
106k views

Bottom to top explanation of the Mahalanobis distance?

I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still ...
13 votes
5 answers
15k views

normal approximation to the binomial distribution: why np>5?

Nearly every text book which discusses the normal approximation to the binomial distribution mentions the rule of thumb that the approximation can be used if $np\geq5$ and $n(1-p)\geq 5$. Some books ...
0 votes
0 answers
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Variance of a symmetric random variable

Let say I have a multivariate random variable $x$ with dimension $s$ which is symmetric at 0 and not necessarily be a multivariate normal distribution. Then what is the variance of $Ax$, where $A$ is ...
1 vote
1 answer
51 views

Justification for conducting independent samples t test on non normal data

I've conducted a study comparing two different teaching strategies in two different sections of the same class; I gave participants in both classes the same 5 tests and wanted to see if test grades in ...
2 votes
1 answer
970 views

When are real limits used for calculating z score?

I'm taking an introductory statistics course. At first, the textbook talks about real limits in the context of continuous variables and frequency distribution table, that is all clear. But on what ...
-1 votes
0 answers
22 views

Why can't the random-varibale $Q_{k-1}$ be computed once we observed $X_1,...,X_k$? and why its distribution becomes $\chi^2_{k-3}$?

We encounter hypotheses $H_{0}$ in which the multinomial probabilities $p_{1}, p_{2}, \ldots, p_{k}$ are not completely specified by the hypothesis $H_{0}$. That is, under $H_{0}$, these probabilities ...
15 votes
2 answers
4k views

Trigonometric operations on standard deviations

Addition, subtraction, multiplication and division of normal random variables are well defined, but what about trigonometric operations? For instance, let us suppose that I'm trying to find the angle ...
0 votes
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27 views

Standard deviation of rates normally distributed

I have a sample of observations that are rates. To calculate their average, I must use the harmonic mean. This is not debatable. The distribution of the sample observations is a normal distribution. I ...
0 votes
0 answers
26 views

What is the probability that the first number chosen is greater than the sum of the next three numbers chosen? [duplicate]

Let $ E_i\sim \mathcal N(20,3), i \in \{1,2,3,4\}.$ What is the chance that $E_1$ is greater than $E_2+E_3+E_4? $ I started by $\mathbb P(E_1>(E_2+E_3+E_4))=\mathbb P(E_1-E_2-E_3-E_4>0).$ What ...
3 votes
3 answers
742 views

Probability of a value x in a Normal distribution is not zero but some value

I'm totally confused about the concept of Probability Distribution Function for continuous sample space.I read that probability of an event in continuous space is zero and it seems logical since we ...
1 vote
1 answer
714 views

Probability to find a modern human with two teeth in different developmental stages

Each teeth growths from the crown to the root. There are different stages previously described to divide this process, as Crown initiation, one half of the crown, crown complete, root one quarter, ...
1 vote
1 answer
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How exactly is OLS derived from assumption of normally distributed residuals?

Ordinary least square solution of linear regression can be derived from the assumption of normally distributed residuals: $$ e_i=y_i-\hat{y_i}\\ e_i\sim N(0, \sigma^2) $$ What I don't quite understand ...
-1 votes
0 answers
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Are the variances in a F-test in Analysis of Variance "nuisance parameters?"

ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate ...
0 votes
0 answers
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Convergence of a function having a big summation at each sample

I have the following function. $$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$ Here, $U_m$ samples are random numbers coming from a Gaussian distribution $$U_m \sim \mathcal{N}(\mu_u, \sigma_u)$$ and ...
0 votes
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Why does the second term of predictive distribution variance vanish as $N$ approaches $\infty$?

The question comes from a statement in section 3.3.2 of Christopher M. Bishop's "Pattern Recognition and Machine Learning", 2006 edition. The related text is excerpted as follows: My ...
1 vote
1 answer
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Can you create continuous data from discrete data?

I am trying to understand the difference between discrete data and continuous data in a real world application. My example is this: Once a day I do pushups until my body is tired. I am counting how ...
0 votes
0 answers
10 views

Are there restrictions on the use of modified Thompson Tau?

I've been running an experiment in which I apply Modified Thompson-Tau to samples drawn from a standard normal distribution, the goal being to understand how $\alpha$ and the sample size [draw_size in ...
0 votes
1 answer
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Is there a way to calculate the CDF for non-normal data? [closed]

I'm aware on how to calculate the Cumulative Distribution Function (CDF) of the Normal distribution. In R, e.g., one can calculate this given a normal distribution by using ...
0 votes
1 answer
24 views

Mass around mode in high dimensional Gaussian

I came across this quote in this paper. The paper is about training a classfier on images with added Gaussian noise $\delta \sim N(x, \sigma^2I)$. The paper states: In high dimension, the Gaussian ...
1 vote
1 answer
23 views

For multivariate normal posterior with improper prior, why posterior is proper only if $n\geq d$

This is related to Gelman's BDA chapter 3 section 5's noninformative prior density for $\mu$. Let $\Sigma$ be fixed positive definite symmetric matrix of size $d$ by $d$. Let $y_1,\dots, y_n$ be iid ...
3 votes
4 answers
9k views

Comparison of 2 distributions

I have 2 distributions, 1 is, as far as im aware, normally distributed. Distribution 1 is the control group. 1st distribution Mean = 0.000002757; Median = 0; StDev = 0.00119307; Number of data ...
1 vote
1 answer
621 views

Generating lists of numbers which sum to points on a normal distribution

In short, I would like to take a number, say N, and generate a list of n numbers which will sum to N with the constraint is that N itself is a number from a nearly normal distribution with, say $\mu \...
1 vote
3 answers
930 views

The principle of getting the error bar of the MLE of the mean of some univariate Gaussian

I'm reading the book Information Theory, Inference and Learning Algorithms. In Section 22.1, the author gives an example of finding the MLE of the mean of an univariate Gaussian, and then obtaining ...
2 votes
2 answers
45 views

Posterior mean of multivariate normal distribution in Murphy's Probabilistic Machine Learning

In the new Probabilistic Machine Learning book by Kevin P. Murphy, to formulate the posterior mean of a multivariate normal distribution, he first defined the likelihood as However, I can't seem to ...
1 vote
1 answer
33 views

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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