Questions tagged [truncated-normal]

The truncated normal distribution is a normal (Gaussian) distribution that as been "cut off" at one or both ends.

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How do I find the standard deviation that results in a specific probability coverage in a truncated normal distribution?

Given a truncated normal distribution $X$ with mean $\mu$, lower limit $a$, and upper limit $b$. How can I pick a standard deviation $\sigma$ such that $P(\mu -x\leq X \leq \mu+x)=y$ for some ...
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Precision issues using truncnorm in R [closed]

I'm facing some precision issues with truncnorm, particularly the ptruncnorm() function in R. Here's an example where I find the cdf up to $q= 2$, where the truncation point is $1.9$, for several ...
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32 views

Uniform Prior on Normal Mean with Known Variance Implies Truncated Normal Posterior?

Let's say I have a uniform prior $\mu \sim \mathcal{U}(a,b)$, a normal likelihood $y|\mu \sim \mathcal{N}(\mu,\sigma^2)$ with known variance $\sigma^2$, and one observation $y$. Is then the posterior $...
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What is E[X|X+Y < z] with X, Y independent Normals?

Let $X\sim N(\mu_X,\sigma_X^2)$ and $Y\sim N(\mu_Y,\sigma_Y^2)$ and $Cov(X,Y)=\sigma_{XY}$. Define $Z=X+Y$. I know that $E[X|Z=z]=\mu_X + \frac{\sigma_X^2+\sigma_{XY}}{\sigma_X^2+2\sigma_{XY}+\...
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How do I calculate $s$ from $\mathbb P(X+Y>u\mid X<s)=q$

Suppose $X,Y$ are (not necessarily independently) normally distributed, how can one calculate the maximal limit $s$ that $X$ may reach such that the probability of the sum $X+Y$ overshooting a given ...
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Integration by sampling from truncated distribution

I'm reading the book Ben Lambert's Bayesian Statistics: problems and answers, which by the way I like. There is a group of problems in "Integration by Sampling" chapter 12. The first integral is $$...
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25 views

Integral and expected value for multivariate distribution

for the last couple of days I have been struggling with a problem and I was hoping to get some help here. I have a function that looks as follows: $$ f(x,y,z)=\begin{cases} 0 & \text{if} \ (x*...
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24 views

d to r conversion for extremely high versus extremely low groups

Context Suppose I am performing a meta-analysis on the relationship between two variables (say depression and life satisfaction). Unfortunately, many studies in the prior literature do not report ...
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Bayesian experimental design choosing the support of the distribution

I am not an expert on these topics so any help is very much appreciated. I'm not even sure if this question is trivial. If so, please let me know. General Setup: Consider the problem posed in this ...
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38 views

Dealing with measurements falling outside of the theoretical range/boundaries of the data

Imagine I am measuring a bounded variable (with a maximum possible value above which the data doesn't make sense) and I end up with the following dataset with my measurements and measurement errors as ...
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307 views

Compute truncated normal distribution with specific mean and variance

I have a simple setting: I simulate demand patterns that are distributed according to a truncated normal distribution with a given mean and variance after truncation. The truncation is from the left ...
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129 views

Linear Mixed Effects Models for Truncated Normal Distribution

I would like to analyze amplitude differences of discrete oscillations that were detected in a time-series using a thresholding method between three different conditions. The number of discrete events ...
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434 views

Calculating the expected value of truncated normal

Using the mills ratio result, let $X \sim N(\mu, \sigma^2)$, then $E(X| X<\alpha) = \mu - \sigma\frac{\phi(\frac{a- \mu}{\sigma})}{\Phi(\frac{a-\mu}{\sigma})}$ However, when calculating it in R. ...
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Variance of squared truncated normal

What is $$ \text{var}\left[v+\frac{1}{2} X \left(\frac{v-\mu}{\sigma}\right)^2 \mid v \leq \mu \right],$$ where $v$ is normally distributed $\mathcal{N}(\mu,\sigma)$ and $X>0$?
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Correlation between normal RV and truncated normal RV

If I have two normally distributed random variables $a\sim\mathcal{N}\left(\mu_a,\sigma_a\right)$ and $b\sim\mathcal{N}\left(\mu_b,\sigma_b\right)$ with correlation $\rho$, is there a closed form for ...
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IRLS for truncated normal GLM

I have data for which responses fall in $y \in [0,\infty)$ for which, it seems, the standard GLMs based on, say, gamma or inverse-Gaussian fail since they don't allow responses with values equal to 0. ...
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How can I test if a permutation hypothesis test produces valid results based on the probability distribution used?

I'm using the Join Count statistics to get insight if there is spatial autocorrelation in a categorical feature. I would like to test if the pseudo p-value returned comes from a valid hypothesis test ...
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How to sample a vector from Multivariate normal with the last element constraint to positive?

I'm doing MCMC simulation and a posterior is hard to sample. Suppose I need to sample a vector $\beta \sim N(M_{\beta} , \Sigma_{\beta})1_{\beta_{K}>0}$, which mean $\beta$ is a vector with length ...
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197 views

Is the truncated normal distribution symmetric?

I am running a Metropolis-Hastings MCMC to find the distribution of a parameter that takes real, positive values. I was considering using the truncated normal distribution, and was wondering if I have ...
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211 views

What is the distribution of min{0,X} when X follows some general normal distribution?

What is the distribution of $\min\{0, X\}$ when $X$ follows some general normal distribution?
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3answers
477 views

Expectation of truncated normal

Suppose I have a bivariate normal $(x,y)\sim \mathcal N(0,\Sigma)$ . Is there an easy formula for the expectation $\mathrm E(x\mid y>0)$ ?
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Can I consider this as a truncated normal distribution?

I have a variable whose maximum value is fixed (maximum number of days in the observation period). In some plots, the histogram of the log-transformed variable peaks near the maximum, but not at the ...
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Bayesian prior and posterior computation for a truncated normal

I have to deal with data in a Bayesian framework, ultimately devising a Gibbs sampler for inferring all my distributions parameters. Specifically, suppose I observe some univariate data distributed ...
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254 views

Estimate parameters from truncated normal sample [duplicate]

I have a question like this, $X \sim N(\mu,\sigma^2)$ with unknown parameters. Now, a sample of size $m$ generated from X, but filter by X < T, i.e., any number larger than T will be ignored and ...
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249 views

Truncated mulitvariate normal: first two moments

Let $X\in \mathbb{R}$ be a univariate random varible for which it holds that $$ X \sim N(\mu,\sigma^2).$$ where $\mu\in \mathbb{R}$ gives the expected value and $\sigma^2>0$ is the variance. If ...
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173 views

Numerical computation of the means and covariance in a truncated bivariate normal distribution

How can I compute the means and covariance of a truncated bivariate normal distribution? I am particularly worried about the case when the truncation occurs very far from the mean. Is there a robust ...
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143 views

Linear combination of truncated normals

I am trying to calculate the following expression: $$ Z = \mathbb{E}\left[\left| \langle \mathbf{a}, u \rangle \right| \right] = \left| \sum_{i=1}^d a_i u_i \right|, \quad \left\| u \right\| = 1 $$ ...
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100 views

How to find the mean of all z scores above a cutoff

I would like to find the mean of all z-scores above a z-score cutoff. For example, what is the mean of all z scores above a z of 1.96 in a distribution N(0,1). (My calculus skills are modest.)
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1answer
351 views

Moments of the truncated normal distribution (univariate) away from the mean

I need to compute the mean and variance of the truncated normal distribution. For simplicity, let us focus on a standard normal, since the general case can be reduced to this. The PDF is given by: $$...
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80 views

Sampling from multivariate normal conditional on a negative minimum

Let $X\sim \mathcal{N}(\mu,\Sigma)$, where $\mu\in\mathbb{R}^n$ and $\Sigma\in\mathbb{R}^{n\times n}$. How can I efficiently sample from $X | {\min{X}\le 0}$? (I.e. from the distribution of $X$ ...
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190 views

Hypothesis test for truncated distributed random variables in Bayesian Regression

As per the recommendation, I am re-framing my questions. I am doing a Bayesian Regression where the parameters are truncated at zero ($0< \beta < \infty$, Assuming prior to follow truncated ...
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247 views

Formular Derivation - Expected Value Truncated Normal Distribution [duplicate]

i want to derive the following expected value formular for $p(x)= N(\mu,\sigma^2)$ (see https://en.wikipedia.org/wiki/Truncated_normal_distribution): $$E[X|a<X<b]=\mu+\sigma\frac{\phi(\alpha)-\...
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Swiss Cheese Distributions

I am curious about a normal distribution with no probability mass in certain regions, sort of like the complement of the truncated normal. In particular, it will have zero mass in a circular region. ...
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1answer
958 views

Conditional distribution of a normal distribution given it is smaller/bigger than another normal distribution

Say I have two independent random variables $X \sim N(u_1, \sigma_1)$ and $Y \sim N(u_2, \sigma_2)$. I want to get the conditional distribution of X given whether X is bigger than Y or not. $P(X|X<...
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Interval of distribution is underrepresented in sample: Is there a way to estimate frequency in ground truth?

I heard about the concepts of censored and truncated data. Now I'm searching for the right concept to express something that is similar to truncated data: As far as I understood, in truncated data ...
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2k views

If a multivariate Gaussian distribution is truncated what will be the new distribution?

I have a covariance matrix and mean values(OMEGA) for a multidimensional Gaussian distribution (3-dim) as follows (respectively), ...
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162 views

Determining mean and SD of a parent normal distribution from a truncated data set

This seems like it should be easier than it is, but I'm stuck trying to determine if a data set I have represents a truncated normal distribution. The hypothesis I'm testing is that the data you see ...
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1answer
161 views

What is the name of this modified Gaussian?

I am reading a paper that defines $$ \mathcal{T}(x, \mu, \sigma) = A \cdot x \cdot \exp \left \{ - \left[ \frac{x-\mu}{\sigma} + \frac{\sigma}{2 \mu} \right]^2 \right \}.$$ It is introduced in ...
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Estimate truncation point of truncated normal given the mean and variance of the distribution

If we know the pre-truncated distribution follows the standard normal distribution and we know the post-truncated mean and variance, is it possible to calculate the truncated point of the truncated ...
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For $y=x+\epsilon$, how can we obtain the SD of $x$ when $y\gt t$

Given that $$ \begin{align} x&\sim N(0,\sigma^2=h^2)\\ \epsilon&\sim N(0, \sigma^2=(1-h^2))\\ y&=x+\epsilon \end{align} $$ Then $y$ is essentially a bivariate normal distribution with $\...
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346 views

How to analyze random variables with non normal distribution

I'm wondering how random variables can be analyzed using parametric methods if the distribution is not normal. For example if a variable Y is normal distributed but I'm interested only on values Y2 ...
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Support of sum of dependent truncated distributions

In case of independent truncated distributions, we can know the support of their sums from their convolutions. How about the dependent counterparts? In my case, I sum 9 dependent truncated ...
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1answer
196 views

How do you sample $\mathcal N(\mu, \sigma^2)$ from a range?

How can we generate a sample in the interval $[a,b]$ based on a Gaussian distribution? If we have a Gaussian random generator, just by mapping the number to the range and pruning (ignoring) the ...
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1answer
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Sampling from Truncated Normal

I am trying to sample from a truncated normal using the standard probability integral transform as described on wikipedia with $$ x = \Phi^{-1} ( \Phi(\alpha) + U*( \Phi(\beta) - \Phi(\alpha)))*\sigma ...
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1answer
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Generate data if parameters for population are known [duplicate]

I know the mean and standard deviation of a population, and its minimum and maximum value. How can I generate each data value from those parameters (using R) with the assumption that the data is ...
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282 views

Partly truncated and not truncated data

In some places 100 km races stop to count time if a certain time limit has been passed, for example 15 hours. It is difficult, respective, not possible to distinguish which finisher or site was ...
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Why is the output weights matrix initialized in a different way as the word embedding matrix in word2vec?

I was plagued by this problem while reading the tensorflow tutorial. There the word embedding is intialized as follows: ...
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1answer
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Fitting truncated distributions using fitdistrplus with a lower bound of zero

I have been working to fit a normal distribution to data that is truncated to only be zero or greater. Given my data, which I have at the bottom, I previously used the following code: ...
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What is the benefit of the truncated normal distribution in initializing weights in a neural network?

When initializing connection weights in a feedforward neural network, it is important to initialize them randomly to avoid any symmetries that the learning algorithm would not be able to break. The ...
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780 views

Truncated Normal distribution: Theoretical mean outside truncation boundaries

I am working with a truncated normal as posterior distribution in a Bayesian estimation problem. Precisely, it is a normal distribution truncated at 0 from below. When calculating the parameters of ...