Statistical Analysis Stack Exchange
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Questions tagged [rayleigh-distribution]

A non-negative continuous probability distribution characterized by one strictly positive parameter.

If a variable $X$ follows a Rayleigh distribution with parameter $\sigma > 0$ then its probability density function is

$$ p(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}, \quad x \geq 0 $$

The Rayleigh distribution appears frequently when characterizing variables that express the sample distance of a symmetric bivariate normal process.

The Rayleigh distribution is a special case of the more general Rice and Weibull distributions. Certain parameterizations of the Rayleigh distribution are related to the Exponential and $\chi^2$ distributions.

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Sampling distribution of the radius of 2D normal distribution

The bivariate normal distribution with mean $\mu$ and covariance matrix $\Sigma$ can be re-written in polar coordinates with radius $r$ and angle $\theta$. My question is: What is the sampling distribution of $\hat{r}$, that is, of the distance from…
caracal
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Estimating variance of center-censored Normal samples

I have normally-distributed processes from which I get small samples (n typically 10-30) that I want to use to estimate variance. But frequently the samples are so close together that we can't measure individual points near the center. I have this…
feetwet
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Is it possible to convert a Rayleigh distribution into a Gaussian distribution?

...and how might we do this? If possible, I am curious if outliers in the Rayleigh distributed data would also remain outliers in the new Gaussian distributed data. Thanks.
Creatron
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Intuition for Rayleigh PDF

We have ground-truth data $\mathbf{x}^* = (0,0)^T \in \mathbb{R}^{2}$. Furthermore, we have $N$ measurements $\mathbf{x}_i \in \mathbb{R}^{2}, i\in \{1,\ldots,N\}$. We calculate the $N$ error vectors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{x}^*$. We…
user137589
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Parameter estimation of a Rayleigh random variable with an offset

I have data that is believed to be Rayleigh distributed (according to some academic papers). However, when I plot the histogram (probability normalized below) it looks like a Rayleigh distribution with an offset. Is there a way to "standardize"…
Daniel V
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Distribution of Extreme Spread for n, sigma

Simple form provided by WHuber: What is the distribution of the diameter of n points in the plane drawn iid from a bivariate Normal distribution? (Diameter is the greatest distance among any pair of the points.) Original long form: Given a Rayleigh…
feetwet
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How to derive the solution of $F_S(x)=P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)$?

[EDIT] I came across a received signal-to-interference-plus-noise-ratio (SINR), $S$, of a wireless communication system as \begin{align*} S = \frac{\phi|h|^2\rho_1}{1+|g|^{2} \rho _{2} }, \tag{1} \end{align*} where $\phi$ is the power allocation…
nashynash
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Derivation of Rayleigh-distributed random variable

I only have a uniform distribution function between [0,1]. And from this distribution, I should generate a sequence of Rayleigh distributed random variable using some software. Anyhow, I was able to finish the problem using the formula from the…
James Mitch
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Find maximum likelihood given Rayleigh probability function

Problem Suppose we use a Gaussian PDF to express the likelihood of light intensity prevalent on Clear, Cloudy, and Eclipse weather. The probability of a certain amount of light value (positive or negative) given the weather is given by the Rayleigh…
Anthony Krivonos
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Finding mean and variance of $Y = \ln{\left(\sum_i X_{i}^{2}\right)}$ for $X_i \sim \mathrm{Rayleigh}(\sigma)$

For some set of $n$ i.i.d. variables $\{X \}$ which are Rayleigh-distributed such that $$ P(X|\sigma) = \frac{X}{\sigma^2}\exp{\left[-\frac{X^2}{2\sigma^2}\right]} $$ I'm interested in anything we can write down analytically about $$ Y =…
CBowman
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Distribution of distance from center of sample group

We have a bivariate normal process where $X, Y \sim N(0, \sigma)$, with no covariance. (For convenience we can assert that $\sigma = 1$, or that we have a good estimate for its value.) What is the distribution of the random variable $R(n) = …
feetwet
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Median of Rayleigh Distribution

I am not sure how to solve the following problem: The probability density function of the Rayleigh distribution is, $\ f(x;α) = \frac{x}{α^2} e^\frac{-x^2}{2α^2}, x ≥ 0, $ where α is the scale parameter of the distribution. Find the median of the…
Joe Ademo
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Variance of the maximum likelihood estimator of Rayleigh Distribution

I want to calculate the variance of the maximum likelihood estimator of a Rayleigh distribution using $N$ observations. The density probability function of this distribution is : $$ f(\sigma,y_i) = \frac{y_i}{\sigma^2}…
Dust009
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Rayleigh distribution with unequal variances

Suppose we have two independent, uncorrelated random variables $X\sim N\left(0,a^2\right)$ and $Y\sim N\left(0,b^2\right)$ (i.e. $X$ and $Y$ are Normally distributed with mean 0 and standard deviations $a$ and $b$ respectively.) How do I find the…
Efficiency
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How are the Rayleigh Distribution and Weibull Distribution related?

I am trying to work out a physically intuitive way of understanding how the Weibull arises. Also according to the Wikipedia entry on the Weibull distribution, there is some how a relation to the Rayleigh distribution. I think it is clear to see how…
Q.P.
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