**3.491 normal-distribution questions.**

$X$ follows a normal distribution $X \text{~} N(\mu, \sigma^2) $.
And there are $n$ samples. Then what is the distribution of $$\frac{1}{n} \sum_i x_i^2$$
I do understand $\frac{\sum_i x_i}{n} \...

Say, there is a $n$-dimension multivariate Gaussian,
$g(x) = N(x:\mu, \Sigma)$ where $\mu$ is $n$-dim mean vector, and $\Sigma$ is $n \times n$-dim covariance matrix.
I would like to calculate "...

Here is the question
"The gross weight of a large bag of landscaping rocks averages 50 pounds with a variance of 25 pounds2. What is the chance that 10 randomly chosen bags weigh a total of at least ...

It's been a long time since basic statistics. I have a financial time-series that exhibits exponential growth.
Before I standardize, do I have to make the time-series stationary?
Before I ...

I have a question about calculation of conditional density of two normal distributions. I have random variables $X|M \sim \text{N}(M,\sigma^2)$ and $M \sim \text{N}(\theta, s^2)$, with conditional ...

Car accidents have a normal distribution between 4:30 and 7 pm. what is the probability of having car accident after 6. I tried assuming the mean was 6, so the value would be zero on the z table, ...

A mobile app I am creating shows a sequence of headlines which when tapped on shows more detailed information. The detailed information can belong to one of several, but small, categories and has a ...

In Bayesian Regression, I am confused how to to get $f*$ and $\sigma*$, given
$$y^∗ \mid \vec{y}\sim\mathcal{N}(f^∗ , σ^∗ )$$
$$
p(y^* \mid \vec{y}) = \int{p(y^* \mid \vec{w}) p(\vec{w} \mid \vec{y})...

This answer notes that if a programming language/libraries provide a procedure that returns random samples from a standard normal distribution, we can generate samples from another normal distribution ...

suppose I have something like a targeting problem, where I specify an angular dispersion in the up and down direction with two gaussian distributions, each having a mean of 0 and a std of 0.3 degrees.
...

Given a $5 \times 2$ dataset $\mathbf{X} =\left( \begin{array}{rr}-0.9&0.2\\2.4&0.7\\-1.4&1.0\\2.9&-0.5\\2.0&-1.0 \end{array} \right)$. Assume that $X\sim N_2(\mu, \Sigma)$.
...

I am confused about the differences in the regression techniques available.
Take for example, linear regression. In this case, we construct a model $y = \beta^Tx + \epsilon$ where $\epsilon \sim N(0,\...

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 ...

Sigmoid transform of Gaussian random variables does not have an easy calculation of covariance.
Specifically, I'm looking for a function transform $$f: \mathbb{R}\rightarrow [0, 1]$$
such that it ...

I have input $X$, which follows a distribution $P(X)$, which is best modeled using a mixture of Gaussians. I also have another random variable $T$, which is also best modeled by a mixture of Gaussians....

We draw $N$ samples, each of size $n$, independently from a Normal $(\mu,\sigma^2)$ distribution.
From the $N$ samples we then choose the 2 samples which have the highest (absolute) Pearson ...

I do not have strong math background, but I am trying to understand Gaussian Processes by example using the lecture Machine learning - Introduction to Gaussian processes by Nando de Freitas. Here is ...

Specifically, suppose $X$ and $Y$ are normal random variables (independent but not necessarily identically distributed). Given any particular $a$, is there a nice formula for $P(\max(X,Y)\leq x)$ or ...

Do all variables in a VAR (Vector Autoregressive model) need to be normally distributed? Or there is no restriction about the distributions of the variables in this model (normal or otherwise)?

This problem is number 4.8 from the Elements of Statistical Learning by Hastie, Tibshirani, and Friedman.
Consider the multivariate Gaussian model $(X|G = k)\sim N(\mu_k,\Sigma)$, where $k$ is one ...

Help me please
Are the following statements true or false? Explain your answer.
Good forecast methods should have normally distributed residuals.
A model with small residuals will give good ...

Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically:
I understand what p( A | B ) where A="I am sick" and ...

Can someone show why the ratio is $\sqrt{\frac{2}{\pi}}$ ?

I'm at a beginner level, so please bear with me.
This is a call center use case. For every week, certain number of calls are received. The average is about 20. This seemed like a Poisson distribution ...

I have a data set where the data, when plotted, is not normal. Log-transforming the data makes it normal. Should confidence intervals for the population mean and hypotheses testing about the ...

If $x$ and $y$ are independent and normally distributed:$$x\sim N(\mu_x,\sigma_x)$$
$$y\sim N(\mu_y,\sigma_y)$$
and $r$ is a random variable with the following relationship to $x$ and $y$
$$r = \sqrt{...

I have a data set of an insect community composition (17 insect species, raw abundance data, sampled 5 times over 60 days within 24 tanks( 4 replicates). There were 3 treatments( free fish, caged fish,...

I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where
$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...

if $M$ is a $m\times n$ constant matrix and $\eta\sim\mathcal{N}(0,I)$, then does $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M\eta\rVert}{\lVert\eta\rVert}\right]$$
exist? Also, let $x\in \...

Gaussian discriminant analysis models learn $P(x|y)$ and then apply Bayes rule to evaluate
$$P(y|x) = \frac{P(x|y)P_{prior}(y)}{\Sigma_{g \in Y} P(x|g) P_{prior}(g) }.$$ Hence, they are generative ...

Most of the introductory stats textbooks, treat the sampling distribution of the mean as a normal distribution when sampling is done without replacement and n/N > 0.1.
They just use of the finite ...

Does anyone know of an approximation for the logarithm of the standard normal CDF for x<0?
I need to implement an algorithm that very quickly calculates it. The straightforward way, of course, is ...

My question is related to the answer in this post.
The definition of the Gaussian copula is easy to understand and the simulation algorithm as well, but I do not see how does the two relate.
My ...

We're trying to estimate the contribution of a device on a performance indicator on the quality of transmission of some signal. The value for performance indicator is assumed to be normally ...

The mean of a Normal distribution is $\theta$ and variance is 1. I know that $\text{E}(X)=\theta$. Then, if I compute the integral I would use to find $\text{E}(X)$ but instead I only take the ...

I have the following problem.
$X_i \overset{IID}{\sim} Normal(\mu, \sigma_1^2) $, $Y_j \overset{IID}{\sim} Normal(\mu, \sigma_2^2), i = 1, \cdots, m, j=1, \cdots n $
Find the MLE for the $\hat{...

Given that for a standard normal variable $Z$,$p(0<z<0.8) =0.2881$
The value of $p($$\vert$$z$$\vert$ $\geq$$0.8)=?$
I already know how to find $p(z$$\geq$$0.8)$ which is equal to $0.21186$.
...

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...

I am currently doing an experiment in gamma ray spectroscopy and trying to find the standard deviation of the Gaussian fit I have made to the data. The issue is matlab provides me with the confidence ...

I've been searching for weeks now but I can't find a proof for the following relationship between the quantiles of the chi-squared distribution and the quantiles of the standard normal distribution:
$$...

My problem is as follows,
Find the maximum likelihood estimator, $\mu^{MLE}$ from $(m+n)$ samples, where $X_1, \cdots, X_m \sim N(\mu, \sigma_1^2), Y_1, \cdots, Y_n \sim N(\mu, \sigma_2^2),$
where ...

I am using the quickpsy package in R. I would like to have a parameter that scales the standard deviation that quickpsy calculates for the cumulative normal distribution function ...

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 ...

See the question I posted on math stackexchange.
https://math.stackexchange.com/questions/2947908/what-does-x-sim-x-mean-in-probability/2947934?noredirect=1#comment6087816_2947934
I just want to ...

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$
but
$$u = \ln(A)-C-k$$
where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant.
I am ...

I am given 3 things:
$Z$ follows a normal distribution $N(0,1)$
$Y=e^{X}$
$X=3-2Z$
What is the moment generation function of $X$ and the $r^{th}$ moment of $Y$ ($E[Y^{r}]$)?
My attempt:
I know ...

Using the probability distribution density functions dbinom, dnorm, etc. and
the corresponding cumulative probability functions pbinom and pnorm, I noticed that the dnorm density values could be ...

in my exercise, $X$ is the size of a tree trunk, and $X$ follows a normal distribution $\mathcal{N}(9,0.4)$, we want to know $P(8.8\le X\le11.2)$
So I though that I could do this: $P(8.8\le X \le11.2)...

Simple question about MVN pdf. I understand the domain to be [0,1]. However, why does scipy.stats.multivariate_normal.pdf output values above this range. E.g.<...

If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...

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