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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18 views

what does an average from a non-normal population mean [duplicate]

i have an elemenary question. If I have a non-normal distribution and get the average does it mean anything? Let's assume the distribution is not known. By "mean anything" I mean can I draw ...
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1answer
29 views

Limiting variance of normal mean

In Casella's Statistical Inference,in Example 10.1.8 on page 470, it says that the limiting variance of normal mean $\bar X_n$, is $\lim_{n\to\infty}\sqrt n\text{Var}\bar X_n=\sigma^2$. However, since ...
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24 views

Understanding the marginal distribution of multivariate normal distribution

I am trying to better understand the multivariate normal distribution. Here I try to refer to the conditional distribution part of wiki also the fifth page of this tutorial. I do not quite ...
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1answer
56 views

What is not normal distribution?

I am running analyses for my dissertation and just got myself in a muddle. I am running descriptive stats to determine whether to use paired t-tests vs Wilcoxon signed rank tests. I've read that it ...
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28 views

Slope analysis where one group is not normally distributed

I am currently doing analysis of five separate drug groups and their changes in levels of prescription over time. Having done linear regression on excel I am comparing the 'steepness' (m) of the ...
2
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0answers
37 views

Distribution ratio of $\bar{y}$ and $S_x$ when $X$ and $Y$ are correlated normally distributed

If we have two correlated $X\sim N(\mu_2,\sigma^2_2)$ and $Y\sim N(\mu_1,\sigma^2_1)$ with correlation $\rho$, then $\bar{y}$ has $N(\mu_1,\frac{\sigma^2_1}n)$ and $S_x$ follows ...
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1answer
29 views

Is my data normal graphical vs. analytical test

I am trying to determine if my data is normal. I am using R. I run the jarque bera test that has a NULL hypothesis of Normality ...
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23 views

Independence of the products of two independent normal variables with a Rademacher variable [on hold]

Let $X$ and $Y$ be two iid $N(0,1)$ random variables, and let $Z$ be a Rademacher variable i.e. $P(Z=1)=P(Z=-1)=0.5$ distributed independently of $X$ and $Y$. Let $U=XZ$ and $V=YZ$. Are $U$ and $V$ ...
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44 views

Mean and Standard deviation [on hold]

The filling machine used by a dairy company to fill 1kg containers of yoghurt produces output which follows a normal distribution with mean 1030g (slightly more than 1kg) and standard deviation 20g. ...
4
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1answer
269 views

Height of a Normal distribution curve

For a Normal distribution 'bell-shaped' curve, one would have thought that the height should have an ideal value. Knowing this value may be one quick indicator to check if the data is normally ...
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1answer
48 views

What are the mean and variance of the ratio of two normal variables, with non-zero means?

If X,Y are normal independent N(a,s), N(b,s') what are means and variances of the ratio X/Y ?
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19 views

Integration of (Gaussian pdf)/r. from 0 to R [on hold]

Integration of exp(-((r-t)/(sigma*sqrt(v)))^2)/r. as this has no defined anti-derivative. I Have tried to approximate the exp() function but that is working to. suggest some other way around. i dont ...
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0answers
22 views

Getting the probability density kernel estimator with R

I am working on a density estimation project and I need to get an estimation of the density as well as an equation for the density estimator (and not the estimate). I am working with kernel ...
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0answers
24 views

Relationship between the parameters of the Normal distribution and parameters in the probit with multiple predictors?

According to A. Agresti (2007, p. 73) in binary probit regression: "The parameters of the normal distribution relate to the parameters in the probit by mean (mu = -alpha/beta) and standard deviation ...
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21 views

Unsatisfied normality distribution and satisfied error of prediction

Sorry if that question is so obvious; I am a newcomer in statistics. I have a set of calculated descriptors that approximate experimental parameters. Because of lack in my knowledge, I encountered ...
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0answers
26 views

Entropy of multivariate gaussian mixture random variable

Short: ${\bf X} \sim N({\bf 0},{\bf I}+{\bf I}_j)$; ${\bf I}_j\in S=\{I_j: I_j$ is diagonal and $ I_j \succeq 0\}, |S|=K$, and $j\sim U(1,K)$. What is $h({\bf X})$? What happens when ...
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1answer
73 views
+50

How to select the best fit without over-fitting data? Modelling a bimodal distribution with N normal functions, etc

I have an obviously bimodal distribution of values, which I seek to fit. The data can be fit well with either 2 normal functions (bimodal) or with 3 normal functions. Additionally, there is a ...
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0answers
19 views

For a normal distribution, find the probability that a measurement is: [closed]

(a) more than two standard deviations above the mean (b) more than two standard deviations below the mean (c) more than 1.67 standard deviations above the mean (d) more than 0.85 standard ...
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1answer
15 views

log multivariate normal differentiation (MLE)

I've come across a lot of explanations of how to differentiate the multivariate normal, but they all appear to skip the step that I'm stuck on. Here's what I've got so far. By logging and removing ...
2
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1answer
42 views

how to get the critical region for a uniformly most powerful test for mean of normal?

I need help in understanding how to construct a uniformly most powerful test using the Neyman-Pearson lemma. Here is an excerpt in my text that I have trouble following: I have no idea how to get ...
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1answer
60 views

How to describe a function of two normal distributed random variables

I consider the generic problem $W(X,Y)=-2\ln(\frac{(X-Y)^2}{2(X^2+Y^2)})$ where $X$ and $Y$ are normally distributed random variables Can I make any statements about the distribution of $W$?
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2answers
63 views

Probability from normal distribution: < vs <=

I want to calculate the probabilities $P\{X < 0.5\}$ and $P\{X \leq 0.5\}$. $X$ is standard normally distributed. From what I have learned density function $\text{df}(x)$ I can get $P(X = x)$ ...
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1answer
43 views

Is the CDF really just a running total of the PMF or am I thinking about this wrong?

When using discrete variables (like the possible outcomes of rolling 2 die {1,2,3,4,5,6,7,8,9,10,11,12} ) is the CDF the same as a sum of the PMF? For instance, take my table below. This is the ...
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14 views

How to run ANOVA on multiple groups of samples, each composed by different variables

I have a $m$ x $n$ matrix, where the $n$ columns are split into multiple classes. If I had only a $1$ x $n$ vector, I would have used ANOVA to evaluate if all subset of columns had the same ...
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36 views

Approximate normality of distribution of counts in contigency table

I seem to be struggling with lack of basic understanding of some important concepts. This is a question to the answer of @Glen_b in this post: Warning in R - Chi-squared approximation may be ...
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1answer
58 views

Construct the likelihood if measurement uncertainties have a Gamma distribution

I want to construct my likelihood. General case: If my data do come from a line of the form $y = mx + b$ and the uncertainties are normally distributed with mean zero and known variance ...
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17 views

Finding the expectaion with respect to a gaussian measure on “half” of $R^n$

I want to calculate the following expectation: $$ \int_{X_\theta} x\phi(x) dx $$ where $\phi$ is the density of a (not necessarily standard) gaussian distribution and $$ X_\theta = \{ x \in ...
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33 views

How to interpret bootstraping output

I have a small dataset which has just 8 elements. I thought I could bootstrap to compare my sample with a normal distribution. I simply want to answer the question: how likely is it that the sample is ...
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49 views

how to construct the likelihood if my errors are not Gaussian

My aim is to study the correlation between 2 parameters knowing that I have measurement errors in both parameters, i.e. I have uncertainties on the independent and dependent parameters. I want to ...
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1answer
21 views

summing standardized or raw variables - does it matter?

I have 2 normal distributed variables, A and B that are correlated with a variable C in a linear regression. Because A and B essentially measure the same latent variable I sum A and B, to increase ...
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2answers
57 views

Probability that the range includes the mean in a sample of $n=4$ from a normal distribution?

If we select one random sample with 4 elements from a normal distribution, and we denote the minimum value among the sample with $a$, and denote the maximum value among the sample with $b$, what is ...
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1answer
27 views

Expectation of a fractional form of chi squared

I have been trying to calculate or find a result for the expectation $$\mathbb{E} \left[ \frac{w^\top D^2 w}{1 + w^\top D w} \right] $$ where $$w \sim \mathcal{N}(0,I_N),$$ and $D$ is a diagonal ...
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0answers
29 views

Expected value of $e^{\alpha \sqrt{t-s} \ Z}$

How can I find the expected value of this: $e^{\alpha \sqrt{t-s} \ Z}$, where Z is a standard normal random variable. I know the moment generating function should help me with this, but I can't ...
2
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1answer
52 views

Normality test for t-test

I ran the Kolmogorov test on a sample and its results showed that the data was significantly drawn from a normally distributed population. Then I assumed that data is suitable for applying t-test. But ...
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3answers
180 views

Transformation of Random Variable - Normal Distribution

Let $X$ be one observation from a $N(0,\sigma^2)$ population . What is the distribution of norm of $X$, i.e., $|X|$ ? My attempt : $$f_X(x;0,\sigma^2)=\frac{1}{\sqrt{2\pi ...
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0answers
7 views

Distribution of magnitude of weighted circularly-symmetric Gaussian random vector

Let $\mathbf{Z} = \{w_nZ_{-n},...,w_nZ_{n}\}$ where $Z_{n\backslash 0}$ is a circularly-symmetric complex Gaussian variable with 0 mean and variance 1, and $Z_{0}$ is a real valued Gaussian variable ...
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0answers
15 views

Transformation Theorem & Piecewise Function

I am a total statistics newbie and I hope that you can help me with the following problem: Let $X \sim \mathcal N\left(\mu, \sigma^2\right)$ be a random variable. Define a new random variable $Y$ as: ...
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1answer
48 views

Generating independent random variables from correlated random variables

I have 2 standard normal, bivariate correlated random variables, $corr \ (X_1, X_2)=\rho$. I want to generate two independent standard normal random variables from these 2. I tried to use what I ...
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3answers
437 views

How does the formula for generating correlated random variables work?

If we have 2 normal, uncorrelated random variables $X_1, X_2$ then we can create 2 correlated random variables with the formula $Y=\rho X_1+ \sqrt{1-\rho^2} X_2$ and then $Y$ will have a correlation ...
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1answer
24 views

Finding expected order statistics from a normal with known parameters [duplicate]

I'm interested in finding the expected value for the kth ordered observation of a normally distributed variable with known standard deviation, mean and n. Could someone let me know the formula for ...
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0answers
12 views

How to show ancillary statisitc of normal random sample?

Let $X_i \sim N(\mu,\sigma^2)$ and $X_i$ are independent. Then how to show that: $$ T = \left(\frac{X_1-\bar{X}}{S},\frac{X_2-\bar{X}}{S},\ldots,\frac{X_n-\bar{X}}{S}\right) $$ $T$ is an ancillary ...
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0answers
26 views

Confidence Interval on the standard deviation [duplicate]

Supposed we have $n = 15$ independent samples $X_1, X_2, ..., X_n$ from distribution $N(\mu, \sigma)$. Sample mean $\bar{X} = 2.4$ and sample variance $\hat{\sigma^2} = 0.55$ What's the 95% ...
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14 views

Joint Gaussian Distribution Question

Drawing a pair of $(x, y)$ from a joint Gaussian distribution with $r$ covariance. Knowing the standard deviations of $x$ and $y$ and knowing $z = x + y$, what is your best guess for $x$?
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1answer
13 views

Factoring a probability distribution containing a latent variable

I distribution which involves 3 parameters, which I'll call (for now) $P(z | y, x)$. However, one of the parameters is a function of another. For instance, let the random variable $y$ be a ...
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25 views

Distribution of estimator

Some help would be appreciated on this one. There is something I can't get around my head. Let's suppose we havfe $ln x$ that is following a Normal distribution of parameter $lnx\rightarrow ...
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0answers
26 views

Bayesian Linear Model Posterior as Sum of Squares?

As part of a homework, I am asked to do the math from the Normal-Inverse Gamma linear regression model. Starting from priors $N(\beta_0, \sigma^2 A)$ and $IG(\alpha_0, \delta_0)$ and with the help ...
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0answers
38 views

Normality and homogeneity

I have performed certain statistical tests (ANOVA, DMRT, t-test, etc.) assuming my data is normal as well as with homogeneous variance. Now my paper is almost accepted in a reputed journal, reviewer ...
0
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0answers
12 views

Product of 3 multivariate normal distribution functions [duplicate]

Is there a relatively simple formula for computing the product of 3 multivariate normal distribution density functions? Where each pdf is defined by: $$ \phi(\mathbf{x}|\mathbf{\mu},\mathbf{\Sigma}) ...
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1answer
39 views

What's the best approach for results of a running race?

I am a student in a good statistics program, but I'm not always the best at picking the tools/process to apply to a problem. To be clear, this is NOT homework, I am asking for a project that I have in ...
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1answer
25 views

Correlation with one variable missing half of its values

Let´s say I want to run a correlation between "eye spherical defect" and height and I want to use only individuals with myopia, whose "spherical defect" goes from 0 to -20 or so. Whereas the ...