# 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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### How prove the distributions of statistic below [closed]

Prove of distribution function of $V=\frac{X_1-\bar{X}}{S}$ where $S^2=\sum_{i=1}^n \left(X_i-\bar{X}\right)^2$
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### Calculating expected value and variance given joint Gaussian distribution

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $X_i: \Omega \rightarrow\mathbb{R}$ random variables for $i=1,2$. $X_1$ and $X_2$ are jointly Gaussian distributed with joint ...
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### Linear Regression Model Variation

I know that Linear regression allows us to model how an outcome variable Y depends on one or more predictor (sometimes called independent variables) $X_{1},X_{2},..,X_{p}$. Equivalently, the linear ...
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### How to “denormalize” a standard deviation?

Lets say I have a vector of variables which have all been standard normalized with $(\mu_n, \sigma_n)$ by doing the operation $\frac{\mathbf{y} - \mu_n}{\sigma_n}$. I then have a model which predicts ...
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### Confidence interval $N(\theta,\theta)$

Let $X_i$ be i.i.d. r.v. with $N(\theta,\theta)$ I calculated $$E[\bar{X_n}] = \theta$$ $$Var[\bar{X_n}] = \theta/n$$ And want to construct a confidence interval $I_{\theta}$ that is centered around ...
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### Expectation of Inverse Normal CDF

Suppose a r.v. $\mu$ is distributed Normal $N(\theta,\sigma^2)$. Is there any way to derive the expectation $\mathbb{E}(\frac{\mu}{\Phi(\mu)})$ where $\Phi$ is the CDF of a standard Normal random ...
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### “Inverse” moment generating function of standard normal distributed random variable

This is just a trivial question maybe but, is the Moment generating function for $X$ the same as for $-X$ for a normally distributed random variable, so $E(e^{tX})=E(e^{-tX})$? If not, what is the ...
Is there a reasonable anti-concentration bound for Gaussian? Let $X\sim\mathcal N(0, \sigma^2)$, can we get $P(|X|>\epsilon)>1-\delta$? Thanks.