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.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Graphically, normal distribution will appear as a bell curve.

enter image description here

However, many other distributions are bell-shaped (such as the Cauchy, Student's $~t-~$, and logistic distributions).

Probability density function: The general formula for the probability density function of the normal distribution is \begin{equation*} P(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-~\frac{(x-\mu)^2}{(2\sigma^2)}},~x\in (-\infty,\infty). \end{equation*}

where

  • $~\mu~$ is the mean or expectation of the distribution (and also its median and mode),
  • $~\sigma~$ is the standard deviation, and
  • $~\sigma ^{2}~$ is the variance.

The case where $~μ = 0~$ and $~σ = 1~$ is called the standard normal distribution. The equation for the standard normal distribution is

\begin{equation*} P(x)=\frac{1}{\sqrt{2\pi}}e^{-~\frac{x^2}{2}},~x\in (-\infty,\infty). \end{equation*}

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

Applications: The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

References:

https://en.wikipedia.org/wiki/Normal_distribution

http://mathworld.wolfram.com/NormalDistribution.html

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Proof of concentration for Gaussian measure using log sobolev inequalities

If $f : \mathbb{R}^{n} \longrightarrow \mathbb{R}$ is $L$-Lipschitz (w.r.t. the $\|\cdot\|_2$ norm), it is a fact that if $x \sim N(0, I)$, then $$ \mathbb{P}( f(x) - \mathbb{E}f(x) \geq t) \leq e^{-t^2/(2L^2)} \:. $$ I'm reading the proof (from…
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proof confidence interval is unbiased for normal sample

I am trying to solve the following question: A $(1−\alpha)100\%$ confidence interval for a parameter, $\theta$, is called unbiased if the expectation of the midpoint of the two endpoints is equal to $\theta$. Consider a sample $\mathbf{Y} =…
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Using importance sampling to simulate the mean of a normal distribution truncated to interval [0,1]

So in these notes it says that importance sampling is: $$\int_F sf(s)ds = \int_G s \frac{f(s)}{g(s)}g(s)ds$$ And then it proceeds to give the following example: In this example, if we draw from $f(x)$, are we effectively drawing from the…
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Convolution with Gaussian function Vanishes

Suppose $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Let $f$ be a function so that $\int |f| \phi dx<\infty$, namely, $f\in L^1(\phi)$. Assume that $(f\ast \phi)(x)=\int f(x-y)\phi(y)dy=0$ for all $x\in \mathbb{R}$. How can one show that $f=0$…
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A covariance matrix of a normal distribution with strictly positive entries is positive definite

This is an intermediate step in a probability homework problem. I have all of it done except for this one step of justification which (I hope!) is true. Let $\Sigma$ be the covariance matrix of an $n$-dimensional Gaussian. Suppose $\forall…
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Distribution of $\sqrt{−2\log(U)}\, \cos(2\pi V)$ without Box-Muller transform

Let $U$ and $V$ be independent random variables that are uniformly distributed on $[0,1]$. Define $$X := \sqrt{−2\log(U)}\, \cos(2\pi V)$$ Prove that $X$ follows normal distribution. I'm attempting to prove that without defining $Y :=…
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Gaussian integrals with cumulative normals

Definitions: Let $$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} $$ be the standard normal probability density function (pdf) and $$ \Phi(x) = \int_{-\infty}^x \phi(t) dt = \frac{1}{2}\left[ 1 + \text{erf}\left(\frac{x}{\sqrt{2}} \right)…
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Calculate the conditional expectation, given two i.i.d. Normal random variables

Suppose $X,Y$ are i.i.d. $N(0,1)$. Then how to calculate the conditional expectation $$E[X \mid X\leq a, X+Y\leq b]$$ Thanks
user6703592
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Conditional PDF on Gaussian random vectors

Suppose the Gaussian random vector $\mathbf{X}\sim\mathcal{N}(\mathbf{\mu_X},\Sigma_\mathbf{X})$ where $$\mathbf{\mu_X}=\begin{bmatrix}1\\5\\2\end{bmatrix}$$ and $$\Sigma_\mathbf{X}=\begin{bmatrix}1&1&0\\1&4&0\\0&0&9\end{bmatrix}$$ I want to…
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Height : Normal distribution

I am looking at the following: The average tallest men live in Netherlands and Montenegro mit $1.83$m=$183$cm. The average shortest men live in Indonesia mit $1.58$m=$158$cm. The standard deviation of the height in Netherlands/Montenegro is…
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problem of normal distribution

The annual income of a head physician is normally distributed. If 25% of the head physician earn less than 180 000 and 25% of them earn more than 320 000, what is the percentage that earn: (a) less than 200 000 (b) between 280 00 and 320…
Anne
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Show that $P(X_2 \leq x \mid X_1 = x)\to0$ when $x\to-\infty$, where $(X_1,X_2)$ is normal with known conditional distribution

Assume that that the conditional distribution of $X_2$ conditionally on $X_1$ is $$f_{X_2\mid X_1}(x_2\mid x_1) = \frac{f_{X_1,X_2}(x_1,x_2)}{f_{X_1}(x_1)} = \frac{1}{\sqrt{ 2 \pi (1- \rho^{2})}} \exp \left( -\frac{(x_{2}-\rho x_{1})^{2}}{{2(1 -…
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Standard Deviation of a Normal Distribution Graph

I am learning about the student t-test. I am struggling, however, to be given a reasonable explanation why the standard deviation of the standard normal distribution curve is 1. It says "The Standard Normal Variable is denoted Z and has mean 0 and…
vik1245
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Find UMVU estimator for $\frac{\mu }{\sigma}$

Let $X_{1},\cdots,X_{n}$ be random samples which has normal distribution $N(\mu,\sigma^{2})$. When $\mu$ and $\sigma^{2}$ are unknown, I want to find UMVU estimator for $\frac{\mu}{\sigma}$. I know that…
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Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the independence between $X+Y$ and $X-Y$ when $X \sim…
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