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.

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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*}


  • $~\mu~$ 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.


6655 questions
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Is the product of two Gaussian random variables also a Gaussian?

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed? Is the answer any different if we know that $X$ and $Y$ are independent?
2 answers

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? If $F$ is the cumulative distribution function…
Chris Taylor
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How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as make sure that the area under the curve was…
9 answers

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^2}{2\sigma^2}}},$$ However, what are $e$…
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Why we consider log likelihood instead of Likelihood in Gaussian Distribution

I am reading Gaussian Distribution from a machine learning book. It states that - We shall determine values for the unknown parameters $\mu$ and $\sigma^2$ in the Gaussian by maximizing the likelihood function. In practice, it is more convenient…
Kaidul Islam
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Gaussian distribution is isotropic?

I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is…
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Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of course, it is closely related to the normal cdf $$\Phi(x) =…
Nate Eldredge
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Affine transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of $\mathbf{x}$. An affine transformation is applied to the…
2 answers

Taking a derivative with respect to a matrix

I'm studying about EM-algorithm and on one point in my reference the author is taking a derivative of a function with respect to a matrix. Could someone explain how does one take the derivative of a function with respect to a matrix...I don't…
5 answers

Product of Two Multivariate Gaussians Distributions

Given two multivariate gaussians distributions, given by mean and covariance, $G_1(x; \mu_1,\Sigma_1)$ and $G_2(x; \mu_2,\Sigma_2)$, what are the formulae to find the product i.e. $p_{G_1}(x) p_{G_2}(x)$ ? And if one was looking to implement this…
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Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$

Since integration is not my strong suit I need some feedback on this, please: Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the expectation value and $\sigma$ is the variance of…
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Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

Let's say I have one point that will be taken randomly from a normal distribution with mean $\mu_1$ and standard deviation $\sigma_1$. Let's say I have another point that is taken much in the same way from another normal distribution with mean…
Justin L.
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Calculate the expected value of $Y=e^X$ where $X \sim N(\mu, \sigma^2)$

I got a problem of calculating $E[e^X]$, where X follows a normal distribution $N(\mu, \sigma^2)$ of mean $\mu$ and standard deviation $\sigma$. I still got no clue how to solve it. Assume $Y=e^X$. Trying to calculate this value directly by…
Jim Raynor
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Characteristic function of a standard normal random variable

The characteristic function of a random variable $X$ is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between this integral and the Fourier transform. For a…
5 answers

How to calculate the Fourier transform of a Gaussian function?

I would like to work out the Fourier transform of the Gaussian function $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$ It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to…
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