# 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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### 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 ...
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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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### 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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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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### 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 ...
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### 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 ...
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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., $a < \mu < b$?
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### 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) ...
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### 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 ...
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