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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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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16 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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18 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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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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20 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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24 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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12 views

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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19 views

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

(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
14 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 ...
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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
60 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
42 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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34 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
48 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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32 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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46 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
55 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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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 ...
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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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179 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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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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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
433 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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11 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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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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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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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 ...
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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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73 views

What is the pdf of a sum of generalized Gamma and normal random variables?

$Z = X + R$ where pdf of $R = \frac{d(\lambda r^d)^n}{r \Gamma(n)} \exp(-\lambda r^d)$ and $X \sim \text{Normal}(0,\sigma^2)$. This is how I began using convolution: $f_Z(z) = f_{X+R}(z) = ...
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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 ...
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175 views

A practical question in simulating real world outcomes

PredictWise aggregated polling data and, for each state, estimated the probability that the Obama or Romney would win. Here is the polling data. The data frame has 51 rows(51 states). the name of this ...
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1answer
50 views

Normal method of moments derivation explanation of Algebra step

In deriving normal estimators using method of moments, why does the below equality hold? $$ \frac{1}{n} \sum X_i^2 - \bar{X}^2 = \frac{1}{n} \sum (X_i - \bar{X})^2 $$ This is from Example 7.2.1 from ...
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27 views

Representing a multivariate normal with a scaled variance

I would like to model an observation to have a multivariate normal distribution but am having some trouble figuring out the linear algebra. So, let us start with a distribution that I know how to ...
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33 views

Probabilistic score vs $L^2$ norm to evaluate Gaussian Mixture Models

There seems to be (at least) two ways that one could evaluate the fit of a Gaussian Mixture Model (GMM) to a data set. First, a probabilistic score, is the log likelihood of a set of points $D$ ...
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1answer
70 views

QQ plot in Python

I generated a qq plot using the following code. I know that qq plot is used to check whether the data is distributed normally or not. My question is what do the x and y axis labels indicate in qq plot ...
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13 views

Prior for a linear transformation matrix: Matrix Normal Distribution

I have been trying to derive some conditional distribution for parameters of a linear transformation (represented as a matrix) and I had a lot of help on this thread yesterday. However, I realised I ...
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24 views

Device Comparision: Correlated or uncorrelated measurements

Background: I want to compare two devices measuring a certain characteristic on a subject. Thereto, each subject is measured once with device A and once with device B. It needs to be assumed that ...
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0answers
19 views

Estimate the conditional distribution of an latent variable?

What techniques might best illuminate the underlying conditional distribution of a latent variable and what information or assumptions would improve that illumination? For example, if we have data ...
6
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54 views

ANOVA: testing assumption of normality for many groups with few samples per group

Assume the following situation: we have a large number (e.g. 20) with small group sized (e.g. n = 3). I noticed that if I generate values from the uniform distribution, the residuals will look ...