The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is often used as a reference against which other distributions are compared.

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Slutsky's Theorem to show convergence to Standard Normal Distribution

We are given $W_n = \frac{\bar{X}-\lambda}{\sqrt{\bar{X}/{n}}}$ and need to show it converges to a standard normal distribution. EDIT: The square root in my original post did not extended over the ...
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20 views

Correlation between normally and non normally distributed variables [on hold]

If one of the variable is normally distributed and the other non normally distributed ... how do u correlate them ? for instance in my case i've two hormones ... one shows a normal distribution while ...
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1answer
16 views

Finding z-scores from z table relating to confidence intervals

I'm having trouble finding the proper $z$ score so that I can find the $99\%$ confidence interval. $\bar{x} = 6.01231$. with an $s$ of $1.96833$ and $n$ of $26$, and I got $2.575$ for ...
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26 views

Proving a “well-known” result regarding the distribution of a normally distributed random variable

In an important project work, I would like to include a "proof" of the following, but have unfortunately been unable to readily compute it myself. I am aware that this is a flaw on my part, but ...
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13 views

Jeffrey's Prior for normal distribution with mean = 0

How would I go about calculating Jeffrey's Prior for a normal distribution with mean = 0, So far I get: But then don't know where to go next. Any help much appreciated
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7 views

Self-study (Expectation Maximization on Bivariate Normal Distribution)

I see this example is also "classic", and I am attempting to understand how to approach it. I have an iid sample drawn from a bivariate normal distribution with mean vector ($\mu_1, \mu_2$) and ...
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12 views

density function of bivariate normal with almost singular correlation matrix [on hold]

Let $X$ be a bivariate normal distribution with mean $[0,0]^T$ and covariance matrix \begin{pmatrix} 1&\rho\\ \rho&1 \end{pmatrix} with $\rho<1.$ I am looking for the behaviour of the ...
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1answer
44 views

How to test a hypothesis about the mean based on an assumed normal distribution?

The entrance onto a major bridge in New York City was engineered to accommodate an average of $3800$ vehicles per hour. However, a random sample of nine observations gives an average of ...
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1answer
41 views

Pointwise convergence of the cdf of normal random variables

For a sequence $X_1, X_2, \dots $, Let $F_n(x)$ denote the cdf of $X_n$. Suppose our sequence is $X_n \sim N(0,n) $ then for all $x$ the point-wise limit of $F_n(x)$ is $\frac{1}{2}$. How would one ...
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2answers
92 views

Relation between sum of Gaussian RVs and Gaussian Mixture

I know that a sum of Gaussians is Gaussian. So, how is a mixture of Gaussians different? I mean, a mixture of Gaussians is just a sum of Gaussians (where each Gaussian is multiplied by the respective ...
2
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1answer
38 views

Degenerate distribution

If $X \, \sim \, \mathcal{N}(m,\sigma^{2})$, I know that $\displaystyle \begin{bmatrix} X \\ X \end{bmatrix}$ is not a Gaussian vector since its entries are not independent. However, what can we say ...
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4 views

Probability with stacked geometric tolerances

I have two datasets for which I know the standard deviations. The data are for printing, where there are certain registration tolerances between different print layers. One dataset is the distance ...
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1answer
36 views

Normality test for large samples

So I working on a programming assignment that uses multiple algorithms to solve the floodit game. I have taken some of my data that I have collected thus far. I did a shapiro test: ...
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2answers
41 views

How to show that a dataset does not contain significant outliers?

I have largish dataset: there is 200 variables and 100 samples. How could I show that the dataset does not contain any significant outliers? All variables have the same unit (millimeters) and have ...
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1answer
14 views

Confidence Interval for 5% Lower Limit

For a normal distribution with a mean of X, and a standard deviation of SD, the 5% lower limit of the population is computed as X-1.645*SD. Meaning, 5% of the population will not reach that level. ...
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8 views

R gplot for normal distribution - add data to graph [migrated]

I'm trying add to my plot some data that will facilitate users. My distribution graph comes from this code: ...
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2answers
37 views

Z-score in the analysis of data

I am being provided z-scores of dependent and independent variables. I was checking if it can analyzed as such as raw data?
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23 views

Correlation between two quadratic forms of Gaussian random vectors

I want to approximately calculate the correlation between two quadratic forms of two Gaussian random vecotrs (of course these are in fact non-Gaussian densities). Does anyone know the derivation of ...
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2answers
39 views

Going from a normal distribution to a standard normal distribution with a change of variable

If $X$ follows a normal distribution with parameters $\mu$ and $\sigma^2$ show that $Z = (X- \mu)/\sigma$ follows a standard normal distribution. This doesn't seem to intuitive to me. We shift $X$ so ...
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1answer
31 views

Best way to define where the data is no longer normally distributed

I thought I'd first give a brief description of what my data is so that it's easier to understand what my problem is. I have a dataset which goes as follows:- binned mass differences of compounds vs ...
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1answer
349 views

Why probability distribution function gives “~.40” probability when it should have been 1.0? [duplicate]

I am following code given here- http://www.bigdataexaminer.com/how-to-implement-these-5-powerful-probability-distributions-in-python/ Under "Normal Distribution" section, the graph peaks at .40 when ...
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28 views

Gaussian random variables [duplicate]

Can some one help me point in the right direction or point to some resources that will help me prove that sum of two jointly distributed Gaussian r.v. with a given correlation coefficient is also a ...
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27 views

Confidence Intervals of a Given Width

I'm working through some questions on confidence intervals. My answer doesn't match the book, but the book's answer was a number I had a few steps before the end. I have ten numbers which are a ...
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1answer
46 views

Normal distribution. Find the average

In a large group of patients, cholesterol level approximates a normal distribution N(μ, σ). Observed that 20% of the members of this group have a cholesterol level of less than 117.7mg/100ml and 8% ...
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43 views

Does this regression diagnostic plot mean my data is invalid, and if so how should I go about fixing it?

I am doing a project on cloud cover and cosmic rays and have undertaken a regression model in R. Above is the regression diagnostic plot and from the QQ plot I can see that the tails are skewed, ...
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48 views

How to deal with 'cut-off' selection bias/sampling bias? (truncated distribution)

In short When measuring an outcome with a normal distribution, but whos mean is below the detection threshold, can you still make statements about differences between populations? Example Say I ...
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26 views

not normally distibuted residuals

I have made an linear regression model using stata. I have made my model diagnostics - predict y, predict (rstudent) residuals. When I control the residuals for normality by a Q-Q plot, it is ...
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49 views

Does stationary data need to be normal?

So I already ran some tests to make my data stationary. Differencing and box-cox transformation in particular. According to the augmented-dickey fuller test, after performing the above mentioned ...
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1answer
26 views

What is the nonlinear transformation assumed by the gaussian (rbf) kernel?

A common kernel choice is the gaussian kernel: $ k(x,x^{'}) = \exp \big( -\frac{1}{2\sigma^2}\| x - x^{'} \|^2 \big)$ This implies a transformation on $x$, and equally on $x^{'}$. What is it?
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25 views

Can near zeros in precision matrix be treated as zeros?

A zero entry in the precision matrix (the inverse of the covariance matrix) means the corresponding variables are indepenent given all the other variables. For real-world data samples, when is an ...
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7 views

Stability precision matrix under small changes in covariance

I am trying to understand how the precision matrix changes under the influence of small changes in the covariance matrix. I have several similar datasets: the differences in standard deviation for the ...
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33 views

How likely is it, that a value belongs to a given distribution?

I'm struggling with this question: I created 100 random data sets and the results are normal distributed. This data will be my null hypothesis. Now I want to check, if an observed value belongs to ...
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Simulating Correlated Normal Random Variables Given Uniforms

I want to simulate $3$ Normal random variables given their expectations, variances, correlations and three independent uniform $(0,1)$ observations. Is my method correct? First, produce three ...
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Comparing two values in a normal distribution after cutting off the tail of the distribution

I'm doing a sports-related analysis about comparing regular-season performance versus playoff performance, in particular which teams tend to do better during the playoffs. Thus, I'm making a "regular ...
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2answers
133 views

Find the mle of $\theta$

This is from Robert Hogg's Introduction to Mathematical Statistics 6th Edition Exercise 6.1.13. The question is: Let $X_{1},X_{2},...,X_{n} $ be a random sample from a distribution on $\mathbb{R}$ ...
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1answer
61 views

Approximate distribution of product of N normal i.i.d.? Special case μ>10σ, σ>0

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $|\mu_X|\geq10\sigma_X$, $\sigma > 0$, looking for: accurate closed form distribution approximation of ...
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1answer
308 views

X, Y are iid from N(0,1). What's the probability that X>2Y

I was thinking, since $X, Y$ are from $N(0,1)$ and they are independent, then $X - 2Y$ has a distribution of $N(0, 5)$. Then $X-2Y > 0$ has probability of $1/2$. The above seems correct to me, ...
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65 views

Approximate distribution of product of N normal i.i.d.? General case

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and NO assumptions about $\mu_X$ and $\sigma_X$, looking for: accurate closed form distribution approximation of ...
9
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1answer
131 views

What is $\mathbb E \lVert X \rVert$ for a multivariate normal $X \sim \mathcal N(\mu, \Sigma)$?

$\DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}$Suppose $X \sim \N(\mu, ...
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1answer
190 views

Approximate distribution of product of N normal i.i.d.? Special case μ≈0

Given $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$, and $\mu_X \approx 0$, looking for: accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic ...
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11 views

Squared Normal RVs Divided by Sum of Squared Normal RVs

Suppose we have $d$ random variables $X_1, X_2, \cdots, X_d$ sampled from the standard Gaussian N(0, 1) i.i.d. What's the distribution of the following identity? $$\frac{X_1^2}{X_1^2 + X_2^2 + ...
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3answers
45 views

Probability for sold items

A salesperson has a probability of 70% to make a sale.
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1answer
50 views

Estimating mean of Normal with unknown variance and then predict the future observation

I am trying to estimate population mean of 9 observations when the variance is unknown. I marginalized the posterior and understand that the t- distribution would give me the distribution of ...
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18 views

Generalized linear model Gaussian distribution Linear Model

Is a generalized linear model with a Gaussian distribution the same as a linear model?
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27 views

How can I show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$? [duplicate]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
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Non parametric test/ANOVA/Parametric test?

My study design is a pre-post test measurements with a control group. My study design is Creatine Kinase being measured before and after exercise, and my subjects are divided into two groups; either a ...
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1answer
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How is $P[X_t\le x_t | X_1,\ldots, X_{t-1}]=P[X_t\le x_t]$ when $X_t\sim WN(0,\sigma^2)$?

In this slide , p.30 , p.31 , it is written that : White noise : $X_t\sim WN(0,\sigma^2)$ i.e., ${\{X_t}\}$ uncorrelated, $\mathbb E[X_t]=0, \mathbb V[X_t] =\sigma^2$ Example : i.i.d noise : ...
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1answer
25 views

Regression Slope and a Bilateral Test

Well, I observe that the standard statistical software package tests regression coefficients if they are statiscally different from zero, that is, not specifically higher nor lower than zero -- a ...
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44 views

Testing normality [duplicate]

How can I (in R) test whether the given data set is normally distributed? I read related questions suggesting shapiro.test. Unfortunately, according to wikipedia, ...
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2answers
102 views

How to compute the PDF of a sum of bernoulli and normal variables analytically?

Can convolution be applied to get a closed form expression for $Z = X + N$ where $X$ is a Bernoulli random variable and $N$ is a zero mean normal random variable independent of $X$?