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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Chance that a vector of observations come from one distribution versus others

My question relates directly to the question at Test to what population an observation came from but I would like to know how to update the information about each possible distribution given ...
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26 views

Asymptotic conditional expectation

Problem Setup Let $\{X^d_1, X^d_2, \cdots, X^d_n\}$ be a $d-$dimensional zero-mean, i.i.d. random variables. Let $S_n^d$ be $$ S^d_n = \frac{\sum_{i=1}^n X_i^d}{\sqrt{n}} $$ Let $Y^d$ be a ...
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1answer
22 views

Transformation from skewed to symmetric distribution

Let us consider a positive valued random variable $X$ which is following a positively skewed probability distribution. Is it possible to a get a function $f$ (one-to-one) for which $f(X)$ follow a ...
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1answer
33 views

Do normal random number generators give a specified sample mean or population mean?

Do normal random number generators like R's rnorm give a specified sample mean or population mean? For example does the mean argument in R's ...
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0answers
12 views

Variance of truncated multivariate Gaussian

Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets. Let $P_S$ be the ...
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1answer
54 views

Bayesian derivation of unbiased maximum likelihood estimator

I was recently reading an old NIPS paper by Bishop and Qazaz where they claim that an unbiased estimator for variance, based on N Gaussian i.i.d. samples with ...
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0answers
16 views

Multivariate log-normal distribution

Let $X=\mu + \Sigma^{1/2}Z$ and $Z\sim \mathcal{N}(0,I)$. Is there a closed form for the distribution of $\exp(X)$?
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0answers
9 views

Hypothesis testing to discriminate between two renewal processes

We have time [0,T] to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
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1answer
17 views

Normality testing on subsets or whole data?

I have data for a continuous dependent variable that I'd like to test for normality. It is a time variable that measures the time of occurrence of an event within 30 seconds. If no event occurs within ...
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1answer
36 views

Conditional expectation for non-gaussian variables

Let $A$, $B$ be two zero-mean random variables. Let the variance be $\sigma^2_A$, $\sigma^2_B$ and let the correlation be $\sigma_{AB}$. Consider the following expression :- $$ ...
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1answer
34 views

Help with intuition about Chi-Squared distribution and its relation to Normal Distribution

At $\alpha = 0.05$, the significance cutoff of a Chi-squared distribution with 1 df is approximately 3.84, and that if we take the square root of 3.84, it's approximately 1.96, which is the 97.5 ...
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1answer
326 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$ (i.e. part of the Black-Scholes call option price): $$\int_{z^*} ...
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15 views

Cumulative distribution discrepancy

The CDF of a normal variable is $P(X \leq x)$, where $X$ is a random variable. This also written as $\Phi (x)$ so if $\Phi (\cdot)$ is the normal CDF, then $\Phi (0)$ is $P(X<0) = 50 \% $ ...
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22 views

quantile of standardized t distribution

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is ...
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0answers
4 views

How can I build a gene regulation network with precision matrix?

Genes are assumed to follow a Multivariate Gaussian distribution, and the precision matrix is asked to estimated. However, after estimation, there are negative values in that matrix, what does that ...
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1answer
29 views

QQ plot problem

Hey guys, I am researching a social science topic on fertility rate per county. However when I finish my linear model and begin to plot my QQ plot, I got a curve that is curving up. Is it a heavy ...
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30 views

show asymtotic normality

Let $x=(x_1,x_2,...,x_n)$ be a sample from a multivariate normal distribution, with mean vector $\mathbf{\mu}$ (n by 1 column vector, all elements equal to $\mu$) and covariance matrix $\Sigma$ ...
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45 views

Central Limit Theorem when the dimension size increases with the sample size

Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$. ...
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28 views

Distribution of sum of squared normals, scalar form

I am working on a problem and hit a wall. I don't need the whole problem answered. just this part. $X_1,...X_n \sim N(\theta, \theta^2)$, what is the distribution of ${\sum_1^nX_i^2}/n$? It's ...
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2answers
77 views

What is $V(X^t)$ for any $t$ when only $E(X)$ and $Var(X)$ are known and $X$ is assumed normal?

Summary I'm trying to calculate $Var(X^t)$ where $t$ is the number of periods using only the following known parameters: $E(X)$ and $Var(X)$. $X$ is a random variable and is the return factor $(1 + ...
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1answer
30 views

Equality in distribution [closed]

I have a problem at hand which involves showing equality of distribution and I have no idea how to proceed and what to show ultimately. Let $$\left(\matrix{U\\V}\right)\sim ...
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19 views

Data does not have a normal distribution but has homogeneity of variance

I'm trying undertake some statistics for my masters thesis but I'm having some problem with my data not being normally distributed. I've essentially got 3 factors, one with 2 levels, one with 3 levels ...
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1answer
20 views

Marginal prior $p(\mu)$ of mean of a normal distribution when both mean and variance are unknown

I read that if the data is normally distributed with mean $\mu$ and variance $\sigma^2$ (both unknown) then to have the joint posterior distribution $p(\mu, \sigma^2 | y)$ in closed form, one has to ...
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16 views

What is the difference between a truncated normal distribution and a half normal distribtion in a Stochastic Frontier Analysis?

I am trying to replicate a SFA where the error term u is assumed to have a cumulative normal distribution function truncated from below at zero. In my opinion, that refers to a truncated normal ...
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36 views

A question on conditional gaussian distribution

The book on Pattern Recognition (by Bishop) begins the section on conditional gaussian by saying: An important property of the multivariate Gaussian distribution is that if two sets of variables ...
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12 views

normal distribution histogram of the residual from a simple regression model?

why residual histogram will follow a standard normal distribution? because independent variable can be different values, I can see all obersavations at one certain X value will follow normal ...
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63 views

Why is the QQ Plot for Normal Distribution a Straight Line?

Pardon my basic question on this, but I could not find why the QQ Plot for normal distribution is a straight line. Also, according to the accepted answer here: Percentile vs quantile vs quartile ...
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1answer
25 views

Pairwise comparison with Bradley-Terry

I am performing a pairwise comparison test for the perceived weight of objects. I want to estimate the difference between each pair, say, A - B. I suspect that the underlying distributions of A, and ...
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38 views

Transformation of data for normality

I'm intended to run a linear regression model (Rain~dBZ) for my data set. I would like to know how to transform non-normal set of "Rain" column in to a normal distribution. I would really appreciate ...
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1answer
353 views

Distribution of a second degree polynomial of a Gaussian random variable

I would like to compute $$P(Y=aX^2+bX+c<0)$$ where $X \sim N(0,\sigma)$. I can do it quite easily using Monte Carlo. However, I've been asked to find the analytical pdf $f_Y(y)$ of $Y$ and then ...
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44 views

How to compute probability

I have a dataset consisting of 4000 observation from each 324 continuous features are extracted. Each observation has been labeled a class. Since each feature from that dataset is continuous, have I ...
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1answer
45 views

Distributions that are fully specified by second order statistics

Apart form the Gaussian distribution are there any other distributions that are fully described by second order statistics?
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64 views

Expectation of rational formula

I have two independent normal random variables $x$ & $y$ that are zero mean and unit variance. $a$ & $b$ are positive. I need to find the mean of $$z=\frac{ax^2y^2}{1 + bx^2}.$$ Any help ...
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Generate Normal distributions from a dataset [closed]

I have data set which is stored as a matrix where each row is an observation (number of observations listed is 4000 ) and each column the feature extracted from that observation (number of features ...
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1answer
30 views

If two stats have same stdev but different distribution will their variance be the same?

So if two distributions are, say, normally distributed and have the same standard deviation, they should have the same variance, right? How about if they aren't both normally distributed?
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Implication of using independent priors for means of joint normally distributed random variables

I am using Bayesian methodology to estimate parameters of joint distribution(Multivariate normal) of random variables $(y_1, y_2) \sim N(\mu, \Sigma)$. I implemented the code for finding the posterior ...
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24 views

How do I check data for normality, then what next…?

I am currently undertaking a hypothesis test for some financial returns data. I'm getting slightly confused on parametric versus non-parametric testing. I am testing a market portfolio of 305 stocks ...
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1answer
37 views

normal approximation to the binomial distribution: why np>5?

Nearly every text book which discusses the normal approximation to the binomial distribution mentions the rule of thumb that the approximation can be used if $np\geq5$ and $n(1-p)\geq 5$. Some books ...
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1answer
9 views

MGF of squared non-standard normal rv

For $Z \sim N(1,1/2)$, find the moment generating function of $W=Z^2$. I tried this: $$M_W(s)=E[e^{sW}]=E[e^{sZ^2}]=E[e^{s(\mu+\sigma x)^2}]=E[e^{s(\mu^2+2\mu\sigma x + ...
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37 views

Estimating a normal distribution from three order statistics

I am interested in predicting a normal distribution, but not sure if this is possible. I do not have information on the mean or standard deviation. However, I know the range of values, let's say ...
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14 views

Cross correlation of gaussian signals with its mean signal gives non-gaussian distributed scores

The following is my question: I have signals that contains noise, they are of the following form see the figure below.. Then I take the mean signal of all these signals (identical in length and ...
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classification for standard normal features

I have an artificial binary classification problem and I know each feature follows a standard normal distribution. For example, we have some standard normal independent features $x_1,x_2,...,x_n$, I ...
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2answers
36 views

Rationale for $E[Z^4]=3$

Given $Z$ is a standard normal random variable with mean 0 and variance 1 ($Z \sim N(0,1)$), could anyone provide an explanation for why $E\left[ Z^4 \right] = 3$? I know that: $$ \begin{aligned} E ...
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9 views

Compare between median/IQR and reference mean/SD

I have a small set of non-normally distributed measurement data (Kolmogorov-Smirnov rejected similarity to a normal distribution) and a reference value from a large population (n=120) of healthy ...
4
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1answer
34 views

ELO rating for non-pairing sport + serious math

I was considering sport disciplines for which there are multiple players at the event but rather than playing against each other, they do stuff, are assigned points and their final position is based ...
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15 views

For a standard normal distribution, are p-values calculated using a $\Phi(-Z)$ or $\Phi(Z)$?

Let $\bar{X}$ denote the value of the sample average for a given set of iid random variables $\lbrace X_i\rbrace_{i=1}^n$. Assume each $X_i \sim \mathcal{N}(0,1)$. According to my econometrics ...
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67 views

Lower-bound for total-variation between two sub-gaussian random variables

Problem setup Let $X$ and $Y$ random variables on the real line with following properties: $X$ and $Y$ are $\sigma^2$ sub-gaussian $E[X] = 0$ and $E[Y] = \Delta$. Question Whats the lower ...
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24 views

Learning matrix with gaussian process

Assume we have a matrix $A$ and that its rows are normally distributed (we assume a gaussian prior for the rows of A). Now, we want to learn the matrix A. The problem I find is in determining the mean ...
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Testing normality of variables for large set of observations [duplicate]

I have hundreds of variables each has more than 100000 observations. My aim is to determine which variables ara normally distributed. Usual tests like Shapiro-Wilk tests are likely to reject normality ...
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1answer
69 views

MLE estimate of $\beta/\sigma$ - Linear regression

I have a question regarding Maximum Likelihood Estimate in linear regression model without intercept. I have a model: $$Y_i=\beta X_i +\epsilon_i, \ \ i=1,...,n$$ where $\epsilon_i$ are i.i.d. $N(0, ...