# Questions tagged [normal-distribution]

This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

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### Diffrence in probability distributions of sepertaed groups

If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and ...
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### In the 1992 presidential election Alaska's 40 election districts averaged 1886 votes per district for President Clinton The standard deviation was 600 [closed]

In the 1992 presidential election, Alaska's 40 election districts averaged 1886 votes per district for President Clinton. The standard deviation was 600. (There are only 40 election districts in ...
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### Proof of Independence of Sample mean and sample variance

This concerns Section 7.8.2 in the book A First Course in Probability by Sheldon Ross 10th Edition. Section 7.8.2 says that : Let $X_1, X_2.\ldots X_n$, be independent normal random variables each ...
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### Does "independent variables X and dependent variable Y are jointly gaussian" means "the residual term has 0 conditional mean"?

I saw the following statement in my lecture note: "The data generation process is $y = x+\epsilon$, whereas in the regression we run y on x so the regression model is $y = \beta_{OLS} x+e$, the ...
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### Compute the probability of multivariate Gaussian distribution

I am reading a proof from a paper: Zhu, Sicheng, Xiao Zhang, and David Evans. "Learning adversarially robust representations via worst-case mutual information maximization." International ...
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### Sampling Normal Distribution; Box-Muller, Inverse Transform, Rejection, Approximations?

I assume $X\sim\mathcal{N}(\mu,\sigma)$ and wish to sample values but I am confused about different approaches and concepts that seem to be relevant for this problem. It appears to me that this ...
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### Exponential distribution from Standard normal distribution

How do we get part (c)? I tried looking for relationships between the chi square distribution and the exponential distribution but couldn't find anything.
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### Conditional density calculation

If we know the density of a vector $x$ is $f(x)$, which could be multivariate standard normal $N(0,I_r)$, and we know the conditional density of $f(y|x)$, which is multivariate normal $N(Ax,I_T)$ with ...
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### A bit of confusion with the Gaussian measure and pushforward by a random vector

Let $d\mu$ be a centered Gaussian measure on $\mathbb{R}^n$ with the covariance matrix $\sigma$ and $X : \mathbb{R}^n \to \mathbb{R}^n$ be any measurable mapping. Or we can simply regard $X$ as a ...
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### Expected value conditioned on highest signal

Let $v\sim N(0,\sigma_v^2)$, let $z, y_1, \dots, y_n\sim N(v, \sigma^2)$ be i.i.d stochastic variables. Calculate $E[v\mid z, z \geq y_i \ \forall i \in \{1,\dots , n\}]$ Said in another way, then ...
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### The correct way to integrate the variational free-energy formula

Lets define the following probability density distributions as: \begin{align} p(θ) &= N(θ; 0,1) \\ q(θ) &= N(θ; μ,σ^2)\\ p(y|θ,x) &= N(y; θx,σ_n^2) \end{align} where $N(x; m,v)$ ...
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I have one basic question maybe is not to hard for you but I am a bit confused. Let our system be like this: \begin{align} X_{k+1} &= A_k X_k + W_k \\ Y_k &= C_k X_k + V_k \end{align} where $... 0 votes 0 answers 21 views ### Overall minimizer of the minus log-likelihood of joint Gaussian distribution Consider the following model. There is a DAG$D_0$whose p nodes correspond to random variables$X_1,...,X_p$: assume that $$X_1,...,X_p \sim N_p (0, \Sigma_0) \text{ with density } f_{ \Sigma_0} (\... • 511 1 vote 0 answers 14 views ### Norm inequality for a linear combination of Gaussian vectors Let us consider a n-dimensional Gaussian blob, i.e. a set of N random vectors \{\boldsymbol{X}^{(j)}\}_{j=1}^N, with n independent components, X_i^{(j)}, and such that X_i^{(j)} \sim \... 0 votes 0 answers 13 views ### Positivity of posterior variance Given a gaussian process with a kernel k, we know the posterior variance after conditioning on data D_n = \{x_1, \cdots, x_n\} is given by$$ k(x,x) - k_n^T(x) K_{nn}^{-1}k_n(x),$$where$k_n(x) = (...
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Suppose that $(X,Y)$ are joint Gaussian with $Cov(X,Y)=\theta_1$, $(Y,Z)$ are joint Gaussian with $Cov(Y,Z)=\theta_2$. Are $(X,Z)$ also joint Gaussian? What is $Cov(X,Z)$?