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

6655 questions
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Characteristic function of the normal distribution

The standard normal distribution $$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$$ has the characteristic function $$\int_{-\infty}^\infty f(x) e^{itx} dx = e^{-\frac{t^2}{2}}$$ and this can be proved by obtaining the moments. However, is there a…
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Proving $E[X^4]=3σ^4$

Given a random variable $X\sim\mathcal N(0,\sigma^2)$, how can we prove that $E[X^4]=3\sigma^4$? I am having trouble even starting with the proof.
foooyoh
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Is $\Phi(q)$ rational for some $q \in \mathbb{Q}^*$, where $\Phi$ is the standard normal cumulative distribution function?

Suppose that we have rational numbers $q_1$, $q_2$ such that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{q_1}e^{-\frac{t^2}{2}} \,\mathrm{d}t=q_2.$$ Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?
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Why do depictions of the normal distribution in textbooks often not look normal?

Here's something I've been wondering for a while. Normal distributions as most of you know look like this (standard normal from -4 to 4): But in textbooks and other serious sources, one often sees images of distributions presented as "normal" but…
Pertinax
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Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian.
Amit
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Integral of a Gaussian process

Let $(\Omega,\Sigma,P)$ be a probability space and $X: [0,\infty) \times \Omega \to \mathbb{R}$ be a Gaussian process (i.e. all finite linear combinations $\sum_i a_i X_{t_i}$ are Gaussian random variables). If the process is continuous, it seems to…
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Calculate $\pi$ from digits of $\pi$

With a random normal distribution $\pi$ can be calculated with help of the PDF (probability density function). The method below apparently shows $\pi$ can be determined with random digits $[0,1,2,3,4,5,6,7,8,9]$. If this is correct and $\pi$ is a…
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Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: $\int_{-\infty}^{\infty}…
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$3\sigma$ rule for multivariate normal distribution

I was wondering if the $3\sigma$ rule that holds for 1D normal distribution also holds for multivariate normal distribution?
shaikh
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Distribution of the sum of squared independent normal random variables.

The sum of squares of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then $X_1^2+X_2^2+\dots+X_k^2\sim\sigma^2\chi^2_k$. How do I go…
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Why don't we allow the canonical Gaussian distribution in infinite dimensional Hilbert space?

I'm looking at Gaussian distributions in infinite-dimensional Hilbert space, and the sources I've seen so far say that the covariance matrix has to be of trace class (i.e. the trace must be finite). Amongst other things this condition rules out the…
Bob Durrant
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What is the Fourier transform of $f(x)=e^{-x^2}$?

I remember there is a special rule for this kind of function, but I can't remember what it was. Does anyone know?
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Standardizing A Random Variable That is Normally Distributed

To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the values such that the expected value is centered…
Mack
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A Mathematical Paradox About Probabilities

So - I am no math genius but I do have shower thoughts. And there is one thought about normal distribution that I just couldn't let go. I converted it into a little story to visualize it a little better. Let's see if it makes sense and if it really…
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How to transform gaussian(normal) distribution to uniform distribution?

I have gaussian distributed numbers with mean 0 and variance 0.2. And I want to transform this distribution to uniform distribution [-3 3]. How can I transform gaussian distribution numbers to uniform distribution?