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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Derivative of multivariate gaussian wrt to covariance

The derivative of the logarithm of a multivariate Gaussian distribution wrt the covariance matrix is: $$ -\frac{1}{2}\Sigma^{-T} + \frac{1}{2}\Sigma^{-T}(x - \mu)(x-\mu)^T\Sigma^{-T} $$ The derivative ...
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1answer
56 views

How to deal with non-normally distributed residuals?

I'm fitting a multiple linear regression model. I've read that the residuals of my regression need to be normally distributed in order for the p and t values to be accurate. Now my residuals (see ...
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10 views

Symmetric Distribution for MCMC Continuous Distribution

I have a sampling distribution $g(X^{'}| X=x)$ such that $$ \log(X^{'})|X=x\sim N(\log(x), \sigma^2)$$ This ensures that our samples are in $(0, \infty)$. Now I would like to use the Metropolis ...
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1answer
19 views

indicator variable - dirac delta or step function

I am trying to solve the following equation, \begin{equation} = \int_{-\infty}^{\infty} \frac{1}{\sqrt{ (2\pi)^{k_{Y}} | \Sigma |}} \cdot \mathrm{exp} \{ -\frac{1}{2} (Y - Xm)^{T} \Sigma^{-1} (Y ...
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22 views

Mulitvariate normal truncated conditional expectation

There is three-variable multivariate normal distribution. Denote 3 variables with $X_1$, $X_2$, $X_3$. Let $\mu_i$ be means, and $\sigma_i^2$ variances of respective variables, and let $\Sigma$ be ...
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26 views

How to derive the conjugate prior for univariate Gaussian distribution(assume both mean and std unknown)?

From google search, it seems Normal-Gamma is the conjugate prior for univariate gaussian. I am wondering if there is a systematic way to derive this ? (or to derive conjugate prior for exponential ...
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1answer
22 views

Find new mean from Normal probability

I have a problem as follows. Life of tyres normally distributed for a specific make. mean=24,000 km and sd= 2500 km. Question is: As a result of improvements in manufacture, the length of life is ...
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1answer
32 views

for normal distributionm, waht is probability x>5y?

It should be a simple statistic question: X,Y ~ Phi(0,1) (normal distribution). What is the probability that X > 5*Y Anyone can teach me how to do it?
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31 views

Regression with Measurement Errors in X and Y

I am trying to find the equation of a line that best fits my data. However, I have errors on the X and Y data points. Here is what my data points look like: ...
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1answer
28 views

N-Dimensioned Normal CDF and Mahalanobis Distance

From Wikipedia's page on "Multivariate Normal Distribution", there's a reference to this PDF: ...
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31 views

How to estimate mean and standard deviation of a normal distribution from noisy data?

I have $n$ observations, $x_i$ following a normal distribution. I would like to estimate $\mu$ and $\sigma$ from my samples. Normally I would simply estimate $\mu=(\sum x_i)/n$ and $\sigma^2=\sum ...
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32 views

Why is this test statistic standard normal? Simple question

My textbook just says that the following test statistic is normal without actually going through the derivation. Here is the problem: Suppose that $X_1...X_n$ are iid RV with each being ...
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58 views

Multivariate Normal Distributions & chi-squared

With a table of data with multiple variables (lets say 2 for ease) and multiple samples how can one calculate the chi-squared test statistic? That is, given $x\sim N(\mu,\Sigma)$, the chi-square ...
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1answer
22 views

Distribution of test statistic under null and alternative

I am currently reading my econometrics notes and there is an example that has really stumped me. The example has an answer with it but I do not understand a few things: Now what I do not understand ...
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1answer
42 views

How to compute the sum of a mixture distribution with another distribution?

I need to find the pdf of x, $f_x(x)$ which is the sum of two random variables $u$ and $w$ and they are independent. I have found the pdf but I am unsure if it is correct or not, the expression is ...
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14 views

How to show via Delta method that the Linear Taylor series expansion of a normal random vector results in NORMAL DISTRIBUTION [duplicate]

How can it be proved using the delta method that the Linear Taylor series expansion of a normal random vector containing independent but NOT identically distributed elements results in a random ...
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105 views
+100

How do I relate the std deviation of the step size, to the stdev of the endpoint of a brownian motion, if the step sizes are multiplied by a function

I know that if I take take a brownian motion of, say, 30 steps of standard deviation 1, then the standard deviation of my endpoint will be sqrt(30). But what if the standard deviation of the 30 steps ...
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Multivariate normal distribution [closed]

Could somebody help me with this problem please? Let X = (X1, X2, X3)′ denote a random vector with distribution N3(μ,Σ),whereμ=(2,1,2)′ and  Σ= 2 1 1 1 3 0. 1 0 1 b) Find the ...
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19 views

Is there a way to check completeness of certain sufficient statistics?

In general, given a p.d.f. or a p.m.f., is there a method to check if a certain statistic is complete? For example, consider a population $N(\theta,1)$ where $\theta$ is unknown and the statistic ...
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1answer
28 views

Determining characteristics of peaks after mclust finite mixture model

I'm working with the mclust package in R (specifically using densityMclust). As output, I have a file with mixing probabilities, variances, and means for each ...
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11 views

Is derivative of a Gaussian Signal also Gaussian? How to find variance of signal that is obtained from differentiation of a Gaussian signal?

Could someone please let me know or give appropriate references for the question I have posed above. My main interest lies in applying Kalman filter for state estimation. The noise on sensor ...
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56 views

How to apply a Gaussian filter to co-ordinate data

I'm working on a project to investigate the correlation of surface finish and face sealing effectiveness. I have a trace of the surface of my seal and the next step is to apply a Gaussian filter to ...
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2answers
36 views

Determine mean from data and variance

I have some data x[i] that differ from the true values by random measurement errors 􏰉cx. Hence one can write x[i] ~ N(mu,cx) My question is: how can I determine ...
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59 views

Distribution of linear combination of OLS regression coefficients

I have a simple linear OLS regression $Y_i = \alpha+ \beta_1 X_{1i} + \beta_2 X_{2i} + e_i$ where $e_i \sim N(0,\sigma)$. I have estimated the regression from the data and obtained estimates for my ...
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1answer
36 views

Which distribution do I get?

Be $X\sim N(\mu,1)$ and $Y\sim Inverse-Gamma(\alpha,\beta)$. For the Inverse-Gamma, I usually use the parameterization which leads to the following probability distribution function for Y: ...
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34 views

Generating random simulations of events [closed]

Whole-edited to make it more simple. Let's assume we have a concrete event: A baseball player's batting average is 0.32. I want to find a random number X, that ...
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15 views

How many samples is needed

I have a function (F) with n parameters (x1,x2,...xn) all the parameters are defined by normal distributions. I would like to know how many times I should sample F to get a reliable distribution. ...
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40 views

Generating a skewed distribution given the median and left and right “$1\sigma$” limits [duplicate]

Edit: I found a solution to the problem, which is at the bottom of the post. I'm going to leave the post as it is in case someone else encounters a similar problem! I was banging my head to a ...
2
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1answer
41 views

MLE for the .95 percentile of the normal distribution

The question is: let $X_1, ..., X_n \sim N(\mu, \sigma^2)$. Let $\tau$ be the .95 percentile, i.e. $P(X<\tau)$ = .95. What is the MLE of $\tau$?_ What I have tried: $P(X<\tau) = P(Z < ...
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1answer
34 views

Poisson or Normal distribution?

In this problem: The respiratory disturbance index (RDI), a measure of sleep disturbance, for a specific population has a mean of 15 (sleep events per hour) and a standard deviation of 10. They ...
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1answer
21 views

Expectation, Variance and Correlation of a bivariate Lognormal distribution

If $Y \sim N(\mu,\sigma^2)$ is normally distributed, then $X=\mathrm{e}^Y$ is lognormally distributed. To get the log-$\mu$ and log-$\sigma$ of this lognormal distribution you calculate $$\sigma^2 = ...
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Estimate The Rate At Which Standard Deviation Scales With An Independent Variable

I have an experiment in which I am taking measurements of a normally distributed variable $Y$, $$Y \sim N(\mu,\sigma)$$ However, previous experiments have provided some evidence that the standard ...
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1answer
105 views

Expectation on higher-order products of normal distributions

I have two normally distributed variables $X_1$ and $X_2$ with mean zero and covariance matrix $\Sigma$. I am interested in trying to calculate the value of $E[X_1^2 X_2^2]$ in terms of the entries of ...
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35 views

How to pool more than two sample means and standard deviations?

I have 4 independent samples from one population, with their respective sample sizes, means and standard deviations. I don't have the raw data, and all of them follow gaussian distributions. Sample ...
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73 views

Does $\sqrt{n}(Y_n-1)$ converge in distribution?

Let $n\in \mathbb{N}$ and consider, for each $n$, independent real valued stochastic variables $Z_{1n}\ldots Z_{nn}$, such that $$ P(Z_{nk}=n)=1-P(Z_{nk}=0)=\frac{1}{n} $$ for $k=1,\ldots,n$. Thus ...
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2answers
134 views

Are there any other distributions whose shapes can be determined by just the mean and the standard deviation?

For the normal distribution, With the mean, and the standard deviation, we know the shape of the specific normal distribution - that is, the center of it and how spread out it is. I wonder if there ...
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13 views

Linear transformation of matrix normal

Let $U$ ($T$ by $m$ matrix) follow a matrix normal distribution with mean matrix $0$ ($T$ by $m$) and covariance matrix $I_T \otimes \Sigma$ where $I_T$ is a $T$ by $T$ identity matrix and $\Sigma$ is ...
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1answer
36 views

Predicting university course marks using historic data of class mean and student's own marks

I would like to predict my course marks for this year based on the data for class mean and my own marks for the past years. What would be a good starting point for a model for such kind of data? ...
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76 views

Why is the squared difference so commonly used?

Very often when I investigate new statistical methods and concepts, I run into the squared difference (or the mean squared error, or a plethora of other epithets). Just as an example, Pearson's r is ...
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A question about the distribution of the log odds ratios in multivariate contingency tables

I know that in $2 \times 2$ tables, one may compute the standard error of the log odds ratio by the formula $$\sqrt{\frac{1}{N_{11}} +\frac{1}{N_{12}}+\frac{1}{N_{21}} +\frac{1}{N_{22}}}$$ where the ...
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1answer
152 views

Advantages of Box-muller over inverse CDF method for simulating Normal distribution?

In order to simulate a normal distribution, from a set of uniform variables, there are several techniques: The box muller; in which one samples two independent uniform distributions $(0,1]$ and ...
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1answer
29 views

How to Map Desired Confidence Interval to a Quantile value

I want to calculate the N% confidence interval for some time-series data set. I have the standard errors for this data series and the error variance of the time-series is assumed to be Gaussian. I ...
2
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2answers
44 views

Standardize non-normal predictors before performing binomial GLMM using mean and sd?

I am planning to predict a binomial variable (1/0, a used point by an animal or point available to an animal in its range) using several continuous, distance-based predictor variables (distance to ...
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Determining whether subsets of a matrix are normally distributed

I have a large image dataset of pixel locations in X & Y and their 'intensity' in Z. I am looping through this matrix in Matlab, creating subsets of the matrix, and I want to test these three ...
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23 views

Approximation of distribution and graphical display in R

The algorithm should display a histogram based on porosity data where every class should have the width 1m representing vertical lithology. Then it would test the data for normality. The algorithm ...
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12 views

Distribution of the probability density values in a multivariate normal [duplicate]

Good day, I'm having trouble with the following question. Suppose i have a random vector following a multivariate normal distribution i. e., $$ \textbf{X}=\{X_1,\dots,X_k\}\sim N_k ( ...
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65 views

Composition of Multivariate Gaussians

This question is teasing my intuition for a moment : $X \tilde{} N(0,S_1)$ $Y|X \tilde{} N(X,S_2)$ Does $Y \tilde{} N(0,S_3)$ with some $S_3 = f(S_1,S_2)$ like (for instance) $S_1+S_2$ ? What I ...
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2answers
67 views

Finding out if your data belongs to normal distribution

Is there a way to find out if your data belongs to one or more (mixture) normal distributions? I probably could calculate what is the standard deviation of my data, but I'm not sure what else to do ...
2
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3answers
122 views

Standard deviation to describe variation in positively skewed data?

I'm wondering how useful the standard deviation is when applied to positively skewed data? The standard deviation implies that 68% of data will lie within one standard deviation of the mean, but ...