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Questions tagged [truncated-normal-distribution]

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

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Probability of sum of normal distributions under threshold, given that each distribution is under a known threshold

Say we have $k$ mutually independent normal random variables $M_1,\ldots,M_k$ where $M_i = \mathcal N(\mu_i, \sigma_i^2)$. We have another normal random variable $R = \mathcal N(\mu, \sigma^2)$ that ...
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8 votes
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Conditional expectation for doubly truncated bivariate normal distribution

The evaluation of the moments of doubly truncated bivariate normal distribution leads to the formulas with a great complexity. It has not been possible to derive explicit formulae for the moments ...
Adrian Daniliuc's user avatar
8 votes
3 answers
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Sampling from $P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$

Is there an efficient algorithm to draw samples $x \sim P(x)$ from the PDF: $$ P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2} $$ where $a\ge0$ is a real parameter, and $m$ a positive integer? Since this is ...
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Efficiency terms in Stochastic Frontier Analysis can be greater than 1?

Let us consider a Cobb-Douglas production function with $Y$ being the output and $X$ being the input (assume for simplicity only one input) and a composite error term: $$ Y = e^{\beta_0}X^{\beta_1}e^{...
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Fitting a truncated normal to data

I consider the normal distribution truncated to the half-interval $[0,\infty)$, $$ P(x) = \sqrt{ \frac{2}{\pi\sigma^{2}} } \frac{1}{\operatorname{erfc}\left( -\frac{\mu}{\sqrt{ 2\sigma^{2} }} \right)} ...
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Adjusting estimated true mean and standard deviations by comparing two truncated normal distributions

I am trying to compare two truncated normal distributions, by estimating unknown mean and standard deviation from truncated distributions. One example is men and women competing in a certain sports ...
Daniel Westergren's user avatar
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Comparing groups that are impacted to different extent by same truncation

Say I have results for a running event, where there is a cutoff time that is the same for all participants. I assume the results are normally or log-normally distributed, but truncated at the cutoff ...
Daniel Westergren's user avatar
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Question on GHK algorithm

After implementing the GHK algorithm, I noticed an unexpected behaviour that lead me to question the validity of the algorithm, but since this is widely used I am sure I must be missing something. ...
adaien's user avatar
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Develop a model for theoretical best performances at different ultra running distances/times

Goal Develop a model for theoretical best performance for running distances from marathon to around 1000 km. Partly to compare the strength of ultrarunning world records, but more importantly, to get ...
Daniel Westergren's user avatar
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Fitting truncated sample to normal distribution with unknown mean & variance

I have data that is somehow truncated. It is a list of log best performances from events, where different events have different cutoff times. How could it be possible to find the unknown mean and ...
Daniel Westergren's user avatar
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On the difference of truncated Gaussian and a new definition

Given a r.v. $X \sim N(0, 1)$, what is the density of $Z = X I(\lvert x \rvert < \lambda)$. I am confused with the truncated Gaussian $Y = X$ if $\lvert X \rvert < \lambda$ otherwise $Y = 0$. My ...
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Correlation of bivariate normal variables with truncated tails

What is the correlation of a bivariate normal distribution after truncating the tails of both variables at $\alpha$ standard deviations? In symbols, what is $$E\left[XY\Big|X| \leq z_{\alpha/2}, |Y| \...
POC's user avatar
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1 answer
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Expected value of Truncated Normal Distribution [duplicate]

For the truncated normal distribution below: $$ {f_X(x; σ, a, b)} = \frac{1}{\sigma}\frac{φ(\frac{x-µ}{σ})}{Φ(\frac{b-µ}{σ})-Φ(\frac{a-µ}{σ})} $$ $$ a = 1; b = ∞; σ = 2 $$ I need to calculate the ...
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Understanding LOO / WAIC for Bayesian models selection

I'm trying to select between two models. 1. has a Truncated Normal likelihood and 2. has a Gamma likelihood. 1. has a much higher WAIC/LOO score but the posterior predictive in 2. (specifically the ...
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How to use Truncated Normal for observation distribution in GLM model?

I'm trying to setup a Bayesian GLM with Truncated Normal, $\mathcal{N}_+(\mu, \sigma, 0)$ truncated at $0$. I want to specify $\mathbf{E} (y\mid x) = ax + b$ but it looks like $\mathbf{E} (y\mid x)$ ...
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How to combine two truncated distributions

We want to combine two truncated distributions to better model one phenomenon. For example, we have a Gaussian distribution, but we want to modify the right hand side tail to make it heavier. So we ...
John Smith's user avatar
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1 answer
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Variance of truncated normal distribution [closed]

I am trying to calculate the variance of a truncated normal distribution, var(X | a < X < b), given the expected value and variance of the unbound variable X. I believe I found the corresponding ...
ajj's user avatar
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2 votes
0 answers
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Computing conditional expectation without integrals or monte carlo sampling

I have gaussian random variables $(Z,Z_k)\sim N(0,\Sigma)$ and $u\sim\text{Uniform}[0,1]$. Given that \begin{align*} y=(1-\boldsymbol{1}\{Z>C\})\boldsymbol{1}\{u<\rho\}+\boldsymbol{1}\{Z>C\}\...
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Mean of a truncated normal distribution [duplicate]

Start with a normal distribution with mean M and standard deviation S. Now exclude all values below the Kth percentile. What is the mean of the remaining 100%-K% of the values as a function of M and S?...
Harvey Motulsky's user avatar
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random number generation of truncated multivariate normal distribution

I want to generate random numbers from truncated multivariate normal distribution specified as follows: $ \begin{bmatrix} Y \\ X \end{bmatrix} \sim N \begin{pmatrix} \begin{bmatrix} \mu_Y \\ \mu_X \...
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1 answer
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Retrieving normal function distribution from truncated normal distribution

I know that a random variable $S_t$ with known statistics $E(S_t)$ and $V(S_t)$ comes from the truncation (sub-sampling) on the interval $[0, +\infty]$ of a random $normal$ variable $S$ with PDF $N_S$ ...
MadMax2048's user avatar
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1 answer
283 views

Calculating functions of truncated and censored normal variables

I am trying to understand why lines (1) and (2) give the same result set.seed(2525); n <- 1e6; X <- rnorm(n) Lines (1) and (2) ...
Manbearpig's user avatar
1 vote
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130 views

Using central limit theorem (CLT) on random variables with known domain, but truncating the Normal(0,1) only on the possible values is more accurate?

For instance, when we have a binomial random variable, we can approximately calculate its CDF using the central limit theorem, but I wonder if we can obtain a more accurate result if we condition the ...
Solano Pillaca Jason Ennio's user avatar
3 votes
1 answer
431 views

Generate a truncated lognormal distribution given mean, variance, lower bound and upper bound?

Basically, I would like to generate a sample of truncated lognormal distribution given mean, variance, lower bound and upper bound. Note that the mean and variance here are the mean and variance of ...
Hongbo W's user avatar
2 votes
0 answers
359 views

Formula for standard deviation of a normal distribution from truncated data [closed]

I have some samples taken from a normal distribution of unknown $\mu$ and $\sigma$, and I know someone took away the top and bottom $p$ percent of the original samples ($p$ is known). Is there a ...
relatively_random's user avatar
1 vote
1 answer
120 views

Conditional Distribution of Normally distributed random variable

Let $x \sim \mathcal{N}\left(0,\sigma^2\right)$ and $y = x+\epsilon$ where $\epsilon\sim \mathcal{N}\left(0,\sigma^2_\epsilon\right)$ and independent of $x$. We know that the conditional distribution ...
Rishabh's user avatar
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2 votes
1 answer
384 views

Fitting truncated normal mixtures in R

I have a vector x, lower_bound < x < upper_bound. I would like to fit a truncated normal mixture distribution to ...
gregorp's user avatar
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1 answer
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How can I generate truncated normal draws from truncated standard normal draws?

Some random variables are transformations of a standard normal. For example, a draw from a normal density with mean $b$ and variance $s^2$ is obtained as$$e = b + s\epsilon$$ where $\epsilon$ is a ...
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create a zero-censored normal distribution from a normal distribution

Say I have a random variable $X\sim \mathcal{N}(0,1)$. Which transformation do I need to apply to $X$ to get a zero-censored normal distribution?
TTT's user avatar
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0 answers
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Sampling from multivariate truncated normal?

I try to implement this procedure. Edit: I report the mathematical steps here: Let $Y \sim TN_d(0,\Sigma, -l, +\infty)$ if $Y = Z | Z \ge -l$ with $Z \sim N_d(0, \Sigma)$ i.e. a multivariate normal ...
paoletinho's user avatar
3 votes
2 answers
285 views

Simulation of a truncated normal distribution over two intervals

Given $X$ a random variable with a normal distribution, what is the best procedure to simulate $X|X\in[a;b]\cup[c;d]$, i.e. we want to simulate the truncated normal distribution only on the intervals $...
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2 votes
2 answers
159 views

Is it possible to calculate or find out what the original distribution was of a dataset? [duplicate]

Is it possible to calculate or find out what the original distribution was of a dataset? For example: I have (part of) a dataset with 800 weights and I know that the original dataset contained 1000 ...
user avatar
2 votes
1 answer
354 views

Why I cannot generate random numbers having a truncated lognormal distribution?

My deduction is: When the distribution is truncated, a normalization factor should be introduced: \begin{equation} g(x) = \frac{C}{x\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{\ln{x}-\mu}{\sigma}\...
Tingchang Yin's user avatar
1 vote
0 answers
153 views

Euclidean Norm normalized Normal Distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anistropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different. ...
rostader's user avatar
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0 answers
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What is truncated gaussian mixture model?

I am interested in the Gaussian mixture model. I read about it and I think I am good with it. However, found that there is something called truncated Gaussian mixture model, which I do not understand. ...
Maryam's user avatar
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1 vote
1 answer
68 views

Calculate mean of $X$ when $X$ and $Y$ are jointly normal and $Y$ is truncated above [duplicate]

Suppose I have two random variable $X$ and $Y$ and they are distributed joint normally and $Y$ is truncated above by constant $c$ $$\begin{pmatrix} X \\ Y \end{pmatrix} = TN\left(\underbrace{\begin{...
user1292919's user avatar
7 votes
1 answer
2k views

Understanding the pdf of a truncated normal distribution

Suppose $\boldsymbol{x} = (x_1, \ldots, x_m)^T$ follows a multivariate normal distribution with 2-sided truncation $a_i \leq x_i \leq b_i$. This is a truncated multivariate normal defined by $TN(\mu, \...
Adrian's user avatar
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0 votes
0 answers
34 views

How to convert a normal random variable to a truncated normal distribution? [duplicate]

Is it possible to transform a normally distributed variable into one that defined by a truncated normal distribution? I am currently using a KL transform to generate Gaussian random fields. I would ...
Trevor's user avatar
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0 votes
1 answer
3k views

How to convert a normal distribution to a truncated normal distribution? [closed]

Is it possible to transform a normally distributed variable into one that defined by a truncated normal distribution? I am currently using a KL transform to generate Gaussian random fields. I would ...
Trevor's user avatar
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0 answers
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Why do I get wildly different answers when computing the mean of a truncated multivariate normal depending on if it is in a for loop or not in R?

I wanted to work with the tmvtnorm package in R to compute the mean of a truncated multivariate normal. In testing the results from this package, I tried to plot ...
thagomizer's user avatar
1 vote
1 answer
62 views

Analytically derive standard deviation of parent normal distribution (with known mean) from moments of the truncated normal distribution

Given $\sigma_t$ and $\mu_t$ of a truncated normal distribution, as well as $\mu_p$ of the parent normal distribution, I would like to analytically compute the standard deviation $\sigma_p$ of the ...
monade's user avatar
  • 519
1 vote
0 answers
74 views

Self Study: Trivariate Normal Expectation with Inequality Condition

I'm reading a paper and found an interesting expectation. I know the result the author found but I can't figure out the intermediary steps because the author provided none. My attempt is getting ...
Panel Noob's user avatar
1 vote
1 answer
829 views

Mean and variance for multivariate truncated normal

Does anyone have a reference for mean and variance of a multivariate normal truncated along a single axis? I.e. $\mathbb{E}[X | x_i > 0]$ and $Var[X | x_i > 0]$, where $X= [x_1,..,x_n] \sim \...
Athere's user avatar
  • 33
0 votes
0 answers
1k views

Normalizing a truncated normal distribution [duplicate]

If I have a normal distribution $X\sim N(\mu, \sigma)$, but I want to truncate it at 0, so that for all values below 0, the density is equal to 0 (like the blue density on the left plot below): The ...
Peter's user avatar
  • 11
1 vote
0 answers
620 views

Finding Confidence Interval for Lower Bounded Truncated Normal Distribution

I am working on finding a confidence interval for data that follows a lower bounded truncated normal distribution (lbtnd) bounded from 0 to $\infty$. I am having difficulty completely understanding ...
Joe's user avatar
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1 vote
1 answer
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If $X \sim N(\mu, \sigma^2)$, then $X|a < X < b \sim \text{truncated normal}$, is it true that $a < \mu < b$?

If $X$ is normal with mean $\mu$, then $X$ in the interval $(a, b)$ is a truncated normal. However, does the mean of $X$ have to lie in the interval $(a, b)$ as well? I.e., $a < \mu < b$?
Adrian's user avatar
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3 votes
1 answer
100 views

Probability of standard normal greater than another standard normal conditional on truncation

$X,Y \sim N(0,1)$ independently. Find $P(Y > 3X | Y > 0)$. My attempt: $$\begin{eqnarray*} P(Y > 3X | Y > 0) &=& P(X < Y/3 | Y > 0) \\ &=& E(1(X < Y/3)| Y > 0) ...
ved's user avatar
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1 vote
1 answer
707 views

Truncated normal distribution without scaling

My understanding of a truncated normal distribution $\mathcal{N}(\mu,\sigma;a,b)$ is that it results from scaling the values of a normal distribution within the bounds $[a; b]$ such that the area ...
monade's user avatar
  • 519
1 vote
2 answers
134 views

Compute Mean of a Clipped Normal Distribution

I am trying to solve this problem and have solved all parts except for the last part. I have tried to research the problem and saw some theory on truncated normal distributions and found this formula ...
Ahsan's user avatar
  • 113
2 votes
0 answers
57 views

Is it Sufficient to Truncate a Left Censored Distribution?

A colleague explained their approach to dealing with left censored data in an analysis, and while I don't think it is the best approach, I am not sure if it is insufficient or not. My colleague has ...
Dave Bapst's user avatar