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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Simple proof that variance of variance estimate is $\sigma^4 \cdot \frac{2}{n-1}$ given normal iid sample

Let $X_1,\dots X_n$ be i.i.d. and $N(\mu,\sigma^2)$ distributed. Let $$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$$ and $$S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i -\overline{X})^2$$ Then I know that ...
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14 views

IQR for Normal Probability Distribution

I was attempting to find the IQR for a normal distribution with mean being 4.5, while the standard deviation. My attempt yielded the result 2.144, while the actual answer provided was 2.1584. ...
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53 views

Self-Study question on Normal Distribution

I was attempting a self-study question on the normal distribution, but I wasn't able to get to the answer. I'm not sure if my method used here is incorrect, and would appreciate some guidance and ...
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22 views

Non-normal observations in regression modelling [duplicate]

I read an article that says the dependent variables in a regression model must be normally distributed. The way i understand it, is that the observations for the regression model must then be normally ...
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1answer
30 views

Continuity correction error when using normal distribution to estimate Poisson distribution

Good morning Everyone, I am doing a self-study exercise which attempts to exemplify a case where the Normal Distribution is used to approximate the Poisson Distribution, since the population mean is ...
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27 views

Finding Percentage Point of a Normal Distribution

Good evening all, I am doing a self-study exercise, but have been quizzed by a part of the question on finding percentage points of a normal distribution. I fully understand the first part of the ...
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1answer
120 views

What does the | in |Z| mean in mathematical expressions for distribution statistics

I am doing some self study on statistics and noticed that in the notes that I was using the $|Z|$ expression as attached in the photos below. I am confused with the "$|$" that is being used. The only ...
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50 views

Meaning of Qqnorm plot in R

I am testing the normality of a sample with R using qqnorm. I obtain this: I understand that the meaning of this plot is that the sample has fat tails. But what is the meaning of the values on the ...
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113 views

Why is the sampling distribution of normal distributed variable automatically also normal distributed

I am currently reading about the standard error. I know about central limit theorem, but I don't understand why, if my variable is normally distributed in the population, the sampling distribution ...
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1answer
29 views

Is it ok to spit non-normal variables in tertiles and put them into multivariate regression models?

I am now reviewing a paper in which the authors decided to predict a DV through linear regression using, beyond other variables, dummy variables obtained from a tertile split of continuous variables, ...
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2answers
43 views

Bivariate Normal Distribution Mean

$X$ and $Y$ have bivariate normal distribution and have joint pdf \begin{equation*} f\left( x,y\right) =a\exp \left( \frac{-1}{2}\omega \right) ,\text{where }% \omega =6x^{2}+12y^{2}-16xy-8x+24 ...
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solve the problem of dichotomous data in winbugs [closed]

i want to write a command in winbugs but i want to write in other method not like the code below so i want to manipulate the problem of dichotomous data. anyone help me thanks alot for(j in ...
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1answer
19 views

Finding the probability of normality via Anderson-Darling, Shapiro-Wilk , and Kolomogrov-Smirov

I have quite a few distributions that were generated by some system, and I am trying to find the probability of accordance to normality for these distributions. So, the probability of accordance to ...
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19 views

Probability of class membership given univariate normal distribution

Assuming a class is well described with a normal distribution of u and s, is it reasonable to calculate the probability of membership as: $Pr(x)=Pr(|x-u|)=2.0*\text{cdf}(|x-u|,s)$? I've briefly ...
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61 views

What is the distribution of this ratio of quadratic forms?

I guess it's F-distribution but I don't know what the solving process it is
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3answers
139 views

Estimating the distribution of a variable

I am trying to estimate (fit) the distribution of a variable. The first step in doing so is to draw a normal probability plot. This is what I have obtained (using R): ...
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61 views

When are $(X, Y)$ jointly normal given that both $X$ and $Y$ are normal?

We know that $X$, $Y$ are normal does not guarantee $(X, Y)$ is jointly normal. A typical example is: $X=Z$, and $Y=ZU$, where $Z$ standard normal, $P(U=1)=P(U=-1)=1/2$, and $Z, U$ are independent. ...
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117 views

Given two independent normal random variables $X$ and $Y$, what is $P(X\leq x\mid X>Y)$?

As the title says, I'm looking for the distribution of $X$ given that $X>Y$.
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repeated measures design between subjects with non normal and small N

I’m examining the effects of a parent-delivered reading intervention on 11 child outcome variables at 3 different time points (pre, post-intervention and follow-up) with a control group. Total sample ...
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The distribution of sample means conditioned on the sample correlation coefficient

Consider the $(X_{i1}, X_{i2}), i=1, \ldots,n $, where $X_{i1}, X_{i2}$ follow bivariate normal distribution with correlation $\rho$. Define the sample means as $\bar X_j= \dfrac{1}{n}\sum_{i=1}^n ...
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34 views

t-test what needs to be normally distributed? [duplicate]

I'm aware that the t-test needs 'normally distributed data'. But take the variable y. When it is plotted without being split by ...
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34 views

Transformation of dependent variable: how to interpret it?

I recently reread some statistics books and noted something weird: They all discuss the assumptions of linear regression and mention the need for a normal distributed dependent variable. In the next ...
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55 views

Simulate the distribution of the median

I have done part(a); now I need to do part(b): I want to simulate the distribution of the median $M$ using R. I don't know how to work it out. And for part(c), do I need to use R as well? This is ...
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40 views

Explaining Gaussian Processes

I am finding it hard to understand Gaussian Processes. Can someone please explain it here in an accessible way? I do understand what Gaussian distribution is but couldn't understand Gaussian ...
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70 views

Multi-peak Gaussian fit in R

I have a large data set composed of several "independent" data frames like this one ...
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62 views

construct the maximum likelihood estimator

Let $a_{1},a_{2},a_{3}$ be independent with a normal(0,1) distribution. Define $X_{1},X_{2},X_{3}$ by $X_{1}=a_{1}$, $X_{2}=\theta X_{1}+a_{2}$ and $X_{3}=\theta X_{2}+a_{3}$ Find the MLE for $\theta$ ...
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49 views

Is the data normally or lognormally distributed?

I'm not sure how to give context to this question. We're to use Excel to analyze data and use log base 10 for each column of data that we analyze, which I'm not sure what they want here. Are they ...
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1answer
240 views

Testing large dataset for normality - how and is it reliable?

I'm examining a part of my dataset containing 46840 double values ranging from 1 to 1690 grouped in two groups. In order to analyze the differences between these groups I started by examining the ...
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44 views

A question between the relation of the 1-alpha quantile and the 1-sqrt(alpha) quantile of the normal distribution

Can I show that $$ P(Z\ge Z_{1-a} - C) > (P(Z\ge Z_{1-\sqrt\alpha}-C/\sqrt2))^2 $$ for every positive number $C$? ($Z_a$ is the quantile $a$ of the standard normal distribution.) What I Tried: In ...
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43 views

Facts about the mathematical difference between Student's t-distribution and normal distribution

I'm looking for some facts (theorems or such) concerning the properties of the Student's t-distribution compared to the normal distribution. More specifically, I understand that for the normal ...
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1answer
49 views

Solve a statistical equation

I got the following exercise where i have to understand the steps theres are done: Since $z_T \sim N(0,T)$: $$\text{Prob}((\mu-0.5\sigma^2)T+\sigma z_T>rT)$$ ...
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101 views

expressing this probability distribution over different variables

I have a likelihood function as follows: $$ P(y|x,w, \phi) = \frac{\phi}{2\pi} \exp ^{-0.5 (y-t(x, w)'\phi (y-t(x,w)) } $$ Here $y$ and $x$ are two observed values. $\phi$ is also some given ...
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167 views

What is the ratio of uniform and normal distribution?

Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it? I found the ratio of two normals with mean zero is ...
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normal distribution mean [duplicate]

Each week, John drives to his mother's house. The amount of time required for the trip varies and is normally distributed. On about 16% of trips, it takes him more than 54 minutes to reach his ...
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131 views

Normal curve probability mean

Each week, John drives to his mother's house. The amount of time required for the trip varies and is normally distributed. On about 16% of trips, it takes him more than 54 minutes to reach his ...
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146 views

How to integrate cdf(ppf(x)-A) for standard normal ppf and cdf

I'm looking for a way of integrating the following formula where ppf() is the percentile point function for the standard normal distribution, cdf() is its inverse, and A is a constant: ...
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56 views

Understanding the effect of each parameter in a Monte Carlo Simulation [duplicate]

I am running a Monte Carlo simulation where I sample from Normal Distributions associated with parameters E11, E22, and GIC to get the plot in red which can be seen in the figure below. The figure ...
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25 views

Approximation of Pr(X > a), with X a multivariate normal rv

Let $X = (X_1, ..., X_p)$ a random variable with a $N(\mu, \Sigma)$ distribution. $$ $$ $$ \Pr(X_1 > a_1, ..., X_p > a_p) \\ =\int_{a_1}^\infty ... \int_{a_p}^\infty (2\pi)^{-p/2} ...
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29 views

Anderson-Darling code test

I'm really rusty at statistics and I'm trying to write some C# code where I feed in a list of numbers and it tells me whether or not the numbers are normally distributed. I generated 50 numbers from ...
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35 views

How to derive the Projected normal distribution

Suppose we have a bivariate normal variable $\mathbf{x}= (x_1, x_2)$ with mean $\mu_1$ and $\mu_2$ and variances $\sigma_1^2$ and $\sigma_2^2$ and correlation $\rho$. I need to obtain the pdf of the ...
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Gaussian Copula Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a Gaussian Copula(or 0 mean gaussian distribution). So ...
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58 views

Avoiding large variances when taking the logs of small values

I have two random variables $(X$ and $Y)$ that are always positive. The assumption I'm making is that their logs follow normal distributions (i.e., $N(\overline{\log(X)},s^2_{\log(X)})$ and ...
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1answer
51 views

What is a good transformation for data that looks like an S on the Q-Q plot? Or a good nonparametric alternative for correlations?

I am trying to do a study to determine if average annual temperature is related to number of cases of a particular disease. I have data for 15 different states over ten years. I have done multiple ...
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47 views

Distribution of the convolution of squared normal and chi-squared variables?

the following problem came up recently while analyzing data. If the random variable X follows a normal distribution and Y follows a $\chi^2_n$ distribution (with n dof), how is $Z = X^2 + Y^2$ ...
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106 views

Probability of Largest Sample Observation

Let $X_1, X_2, X_3$ be three random variable following a normal distribution $N(6,4)$. What is the probability that the largest observation exceeds 8? Hint: $Y = \max(X_1, X_2, X_3)$. Here ...
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146 views

Multiplying Gaussian distributions of different dimensions

The multiplication of multivariate Gaussian distributions defined over some parameter vector of a given dimension can be achieved by the following. Assuming that the Gaussian is parametrized by the ...
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206 views

Determining the probability of $X_2 \ge X_1$ given they have different probability functions

Suppose that I have a random variable $X_1$ which is normally distributed, and a random variable $X_2$ having the density function shown in the figure below. How would I determine ${\rm P}(X_1 \le ...
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Measuring distribution of growth rates

I have monthly sales data for 50,000 companies. Unfortunately, I have very limited statistics training. I suspect/hope that by some measure of growth these companies would fit into a standard normal ...
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1answer
95 views

Difference in two Normal Distributions?

I am unsure what I should do if I have two Normally Distributed variables with known parameters and I want to find the probability that one of these variables is greater than the other. Should I use ...
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1answer
30 views

Probability with Z-Score

Suppose that we model the distribution of IQ scores in the general population as a normal random variable with mean 100 and standard deviation 15. Find the probability that a randomly selected ...