**3.282 normal-distribution questions.**

Say we have a univariate Gaussian distribution $p(x)=\mathcal N(x|\mu, \sigma^{2})$. Then suppose we have a data set of observations $X = \{x_1, x_2, \cdots ,x_N\}^{T}$ that are drawn independently ...

Let $Y_1,Y_2,\cdots,Y_n$ be independently distributed random variables such that $Y_i\sim\mathcal N(\alpha+\beta x_i,\sigma^2)$ for all $i=1,\cdots,n$. If $\hat\alpha$ and $\hat\beta$ be the least ...

I am working through the "Math for Quantitative Finance" course on brilliant.org. The following question was given as an example:
An investor wishes to invest $700.
There are two independent ...

I know there are several questions (here and on other websites) regarding the comparison of values, but I am still lost with the data I have and the analyses I should conduct.
I have just 18 values (...

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be ...

I am unsure about how to analyze my data as they are quite non-normally distributed, which causes model residuals to be also very non-normally distributed. I have seen there are various threads on non-...

Some values have a normal distribution with mean .0276. What standard deviation is required so that 98% of values are between .0275 and .0278?
What I'm confused with is how to calculate the standard ...

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from a $\mathcal{N}(\theta,1)$ population where $\theta>0$.
I am trying to compare the estimators $T=\bar{X}\mathbf1_{\bar{X}>0}$ and $\...

i need critical value for $ t_{249}(0,975) $. I can generate in R in easy way, but in school we will have just tables and in tables are only critical values for degrees of freedom = 100,200,500. How ...

I am studying associated variables for applying them on some proofs of GARCH(1,1).
I found a paper of Newman and Wright (1981), where a formula which I need is given in a theorem. I need help at the ...

I know that two normal distributions when added will give a normal distribution, and through this I can arrive at the standard normal.
However, I need to prove $X$ and $Y$ give a standard normal $Z$ ...

Do anyone know what is Inverse Cumulative Distribution Function of Normal Distribution formula or equation? I am looking at Google but did not find any good answer.

I was reading a research paper about climate change and in it the authors say that, in order to establish the significance of a shift in the mean of a standard normal distribution at the $5\%$ ...

Let's assume I have a test subject of $n$ students. If I already know the mean and sd values how can I find how many students have a value greater than $y$?
For reference let mean = $10$, sd = $5$, $...

I have a question regarding the normality of predictors. I have 100,000 observations in my data. The problem I am analysing is a classification problem so 5% of the data is assigned to class 1, 95,000 ...

I know that if the quantile plot between two random variables is a straight line, then the cumuluative distribution functions of the two are likely linear transforms.
However, after trying to do a Q-...

This question is about the assignment on my ML course..
I have been given two continuous data in a normal distribution and predict the values of both for class labels(m/f) in 2 steps:
build a ...

When I try to solve a problem of a personal research, I get the following integral:
$$E\left(X\sqrt{(\alpha X)^{2}+4}\right)=\displaystyle{\int\limits_{-\infty}^{+\infty}\frac{x\sqrt{(\alpha x)^{2}+4}...

The problem is: randomly generate n positive integers that sum up to a fixed value sum and follow a binomial distribution that approximate a normal distribution.
How would I go about it?
--
...

I'm using R-package "pracma" to calculate a triple integral.
This is my code:
...

I am analyzing satisfaction levels of PhD scholars in a particular region.
We got 7 independent variables from literature review and constructed 45 questionnaires accordingly: for example, 5 ...

According to this page on wikipedia, if $X\sim N(\mu,\Sigma)$ with $\mu\in\mathbb R^2$ and $\Sigma\in\mathbb R^{2\times2}$ then we have $\textrm{Cov}(X_1,X_2)=0\implies X_1\perp\!\!\!\perp X_2$. Is ...

I'm watching a pool match and at the start of the first game the person next to me says
"I think the chance of player A winning the first game is $.700$. If he wins that I think the chance of him ...

I'm trying to split a total sales forecast for (e.g.) 5 shops for April based on March results.
Original budget (B1) prepared was in January, and an updated forecast (F1) was prepared in March. ...

I have my AP Stats exam tomorrow and had this question.
I apologize if it is too simplistic for this forum.
I'm pretty sure the sample's distribution is nearly normal, but not 100%, can someone ...

Why does it say data should be normally distributed for statistical analysis when different test follow its own distribution (i.e. t, Z, F)?
What does normality have to do with this?

I tried to obtain the score vector (1st derivative of density function w.r.t. parameters) of multivariate normal distribution.
Density function of multivariate normal distribution:
\begin{align*}
f(x)...

Considering the following random vectors:
\begin{align}
\textbf{h} &= [h_{1}, h_{2}, \ldots, h_{M}]^{T} \sim \mathcal{CN}\left(\textbf{0}_{M},d\textbf{I}_{M \times M}\right), \\[8pt]
\textbf{w} &...

If $\mathbf{x} \sim N(\mathbf{0,I})$ then $\mathbf{AA}^T$ is the covariance matrix of $\mathbf{y} = \mathbf{Ax}$, but what does $\mathbf{A}^T \mathbf{A}$ represent?
In some places I have seen ...

Apologies if this is a really simple question; I'm sure if only I knew what to google I'd be able to find the answer myself, but it's been driving me mad.
I have two datasets with approximately ...

Suppose
$$
\mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e} \, ,
\\
\mathbf{e} \sim \mathcal{N}(0,\mathbf{I}_P) \, .
$$
We know that $\mathbf{\hat{b}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \...

Assume the following situation:
we have a large number (e.g. 20) with small group sized (e.g. n = 3). I noticed that if I generate values from the uniform distribution, the residuals will look ...

I have the following iid. variables $X_1,..,X_n,Y_1,..,Y_m$ with distribution $X_i\sim N(\mu_1,\sigma_1^2), Y_j\sim N(\mu_2,\sigma_2^2)$. How do I find the minimal sufficient statistic for $(\mu_1,\...

A machine measures the height of some plants. In the context of industrial quality procedures, we repeat the same measurement Hi 10 times and expect the ...

In the canonical form multivariate Gaussian pdf $p(x|\eta,\Lambda) = \exp \Bigl\{a+\eta^Tx-\frac{1}{2}x^T\Lambda x \Bigr\}$, where $\Lambda = \Sigma^{-1},\eta=\Sigma^{-1}\mu,a=-\frac{1}{2}\left(\log 2\...

Imagine I have some process with mean $\mu$ and variance $\sigma^2$, which are both known empirically. If I sample from this process 1000 times, what's the probability that the mean of those 1000 ...

I am sampling a normally distributed noise signal (digital recording) with some arbitrary mean and standard deviation, where a single sample is received each second (1 sample / 1 sec).
I am ...

I have a problem as follows.
Life of tyres normally distributed for a specific make. mean=24,000 km and sd= 2500 km.
Question is: As a result of improvements in manufacture, the length of life is ...

Given two multivariate Gaussian distributions $P \equiv \mathcal{N}(\mu_p, \Sigma_p)$ and $Q \equiv \mathcal{N}(\mu_q, \Sigma_q)$, I am trying to calculate the Jensen-Shannon divergence between them.
...

Is there a conjugate prior distribution in the model $\text{N}(\theta, \sigma^2=\theta^2)$, that is a normal distribution where the mean and standard deviation are equal ($\theta>0$)? (This ...

I've done some experiments to understand the influence of the dimension of the latent space in a VAE, and it seems that the higher the space, the harder it is to generate realistic images. I might ...

In Gaussian random vector,the correlation of two random variables is always between -1 and +1.
How to check this fact by application of Cauchy- Schwartz inequality which states that
$E(XY)â‰¤(EX^2)^{...

I have been trying to find an expression for the covariance of two normally distributed variables X and Y if cov(x,y)=c then cov(x,xy)=?
I would greatly appreciate any help. Probably it must be ...

As we know, when the Poisson rate $\lambda$ is large, we can approximate the Poisson distribution as a normal distribution with equal mean and variance.
Is there a conjugate family for this normal ...

Suppose I have a mixture of finitely many Gaussians with known weights, means, and standard deviations. The means are not equal. The mean and standard deviation of the mixture can be calculated, of ...

it's well known that the scale mixture of normal distributions is equivalent to a Student t model, that is
$$
t_{(v)}(x|\mu,\sigma^2)=\int_0^\infty N(x|\mu,\sigma^2/\lambda)\times G(\lambda|v/2,v/2)d\...

I have data modeled as a mixture of two Gaussian distributions. The data is "clipped" i.e., there is data only for values greater than a threshold $t$, even though it is feasible for data to exist in ...

I have a data with two variables - one variable is quantitative, and one is nominal (4 categories). I ran the normality tests and it turned out that this quantitative variable is not normally ...

How can I generate k bivariate normal random variables with
mean=0
sigma=1 and
correlation=rho in R?

I've been solving this book "40 puzzles and problems in probability and mathematical statistics" and I came across this question of mixing RVs vs. mixing distributions.
Consider a RV, B, that comes, ...

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