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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Normal and Uniform Distribution, calculate $P(Y>X\mid X=x)$

Let $X$ and $Y$ be independent random variables distributed as $X \sim N(0,1)$ and $Y \sim \operatorname{Unif}(0,1)$. (a) Find $P(Y > X\mid X = x)$. (b) Use your answer in part (a) to compute $P(Y > X)$ I'm not sure where to start... I think the…
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Find a and b such that $ P(\sigma_*^2 < a\sigma ^2) =0.025$ and $ P(\sigma_*^2 > b\sigma ^2) =0.025$

Let $X_1, X_2 ... X_n \text{ ~ } N(\mu, \sigma^{2})$ be identically distributed and consider the estimate $$ \sigma_* ^{2} = \frac{1}{n} \sum\limits_{i=1}^n (X_i - \mu)^2 $$ for the variance. Find number $a$ and $b \in \mathbb{R} $ such that $…
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Blitzstein: Mode of a Log-Normal distribution

In Blitzsteins book "Intro to probability" he asks the following question in chapter 6: Let $Y$ be Log-Normal with parameter $\mu$ and $\sigma^2$. So $Y=e^X$ with $X\sim N(\mu, \sigma^2)$. Evaluate and explain whether or not each of the following…
Semoi
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Find the PDF of $\max(R_1,R_2)/\min(R_1,R_2)$ for $(R_1,R_2)$ i.i.d. standard normal

$R_1, R_2$ are two independent continuous random variables, they all satisfied in the following normally distributed equation. $$f_1(x)=f_2(x)=\frac{1}{\sqrt{2\pi}}e^\frac{-x^2}{2}$$ The problem is to find $R_3=\frac{max(R_1, R_2)}{min(R_1,…
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Can a 2 paired sample sets of size 15 (30 readings in total) be assumed to have normal distribution when finding confidence intervals?

I've seen that for large sample sizes (about 30 or more) the distribution tends to a normal distribution, but what if I have 2 sets of 15 which are paired? Would that count as a normal distribution, or would I need to use a t-distribution to…
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The distribution of xtΣx

For the multivariate gaussian distribution $x$, what is the distribution of $x^T\Sigma^{-1}x$?
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Transforming one multivariate Gaussian to another

Suppose we are given a random vector $\mathbf x \in \mathbf{R}^n$ that has a multivariate normal distribution $\mathbf x \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$, and we are also given a positive semidefinite matrix $\mathbf \Sigma_1\in \mathbf…
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log concavity preservation on a difference between two gaussian cumulative distribution function?

Suppose that I have a random Gaussian variable (a vector), $x$, that has zero mean and covariance matrix as $\Sigma$. Suppose that its CDF is denoted by $F(x)$. We know that $F(x)$ is log concave. But I wonder if the following function is also…
StellaLee
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Finding $ E ( X Y \mid X + Y = w ) $ when $ X $ and $ Y $ are i.i.d $\mathcal N ( 0 , 1 ) $ variables

$ X $ and $ Y $ are both independent and identically distributed random variables with normal distributions $ \mathcal N ( 0 , 1 ) $. What is $ E ( X Y \mid X + Y = w ) $? I know this means that $ W=X+Y $ must be normally distributed as well with…
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How do I combine multiple noisy measurement?

Assuming there's a variable I want to measure, but I have only very noisy instrument to do so. So I want to take multiple measurements so that I have a better chance to recover the state of the variable. Hopefully, with each measurement, my…
xing_yu
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Approximation of Exponentially and Normally Distributed Probabilities

PROBLEM A company uses a portable high-intensity flashlight. Batteries and bulbs burn out quickly. The lifetime of batteries has Exponential Distribution with mean $10$ hours. The bulbs have lifetimes that are Normally Distributed with mean $32$ and…
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Approximate probability mass function into normal distribution

I have an array of values (mass function) Px and I want to approximate a normal distribution function (in Matlab) from them. I can plot the mass function using bar(Px) and I would like to plot normal distribution graph given these data, too. How do…
Smajl
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How to normalize this exponentially distributed data?

I have this histogram of data...what is the most proper way to prepare it for consumption in a neural network? I know how to normalize/standardize other types of data, but I'm wondering what to do with this kind of distribution.
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QQ plot and visual analysis based on sample distribution

When would you use a Box Plot, a Histogram, or a QQPlot to graphically summarize a SAMPLE of numbers? Interpretation: a sample of numbers: like a dataset that only contains numbers but not discrete values such as categories. OR: It means whether the…
Shane Gurtin
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gaussian lipschitz concentration about the $p$-th moment

Given $x \sim N(0, I)$ and $f : \mathbb{R}^{n} \longrightarrow \mathbb{R}$ which is $L$-Lipschitz, we have that $f - \mathbb{E}f$ is a sub-Gaussian random variable, specifically that $$ \| f(x) - \mathbb{E} f(x) \|_{\psi_2} \leq C L \:. $$ I want…
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