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Questions tagged [rayleigh-distribution]

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

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Method-of-moment of n IID random variables

The method-of-moment of $\sigma$ for the following pdf is $$ \text{pdf}(x,\sigma) = \frac{x}{\sigma^2}\exp(-\frac{1}{2}\frac{x^2}{\sigma^2}) $$ $$ E[x] = \int_{0}^{\infty}\frac{x^2}{\sigma^2}\exp(-\...
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Denoising data with Rayleigh distribution assumption

Assume I have 3 random variables, $X$ (matrix of pixels of the noised images),$Y$ (matrix of pixels of the denoised unknown image),$Z$ (matrix of pixels of a noise with Rayleigh distribution), related ...
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Sampling the max among $N$ samples from the Rayleigh distribution

I've read on the internet that the pdf of the sample max, $X$, from among $N$ i.i.d. samples from a distribution with pdf $f(x)$ and cdf $F(x)$ is given by $$ p(X) = N f(x) F(x)^{N-1}. $$ I'm ...
Tor's user avatar
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Hypothesis tests for Rayleigh variables

Given samples from two Rayleigh-distributed random variables with unknown parameters, $X \sim R(\sigma_x), Y \sim R(\sigma_y)$, what tests can we use to determine if and to what extent their ...
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Bivariate normal covering circles and ellipses

I am looking at covering circles for cartesian coordinates given by independent bivariate random variables $X, Y \sim N(0, \sigma)$. The radius of a circle that will cover proportion p of these ...
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CDF for squared sum of Rayleigh random variables

In short, I am looking to estimate the distribution of $ \eta = \sum_{i=1}^N (X_i - z_i)^2$, for each $X_i \sim \text{Rayleigh}(1)$ and constants $z_i$. If $X_i$ were Gaussian, then this could be ...
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Adjusting confidence interval of estimator by efficiency

Summary: If we have an unbiased MLE $\widehat{\sigma_1}$ of an exponential distribution parameter, and the confidence intervals for its estimates are given by the $\chi^2$ distribution; and we find ...
feetwet's user avatar
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Computing sum squared distances without computing center

Given an even number of sample points in a plane, I want to compute the sum of squared distances from the sample center as part of estimating the Rayleigh parameter. One way of doing it is to compute ...
feetwet's user avatar
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Expected value of Rayleigh quotient, non-centered Gaussian vector

Let $X \sim \mathcal{N}\left(\mu, \Sigma \right)$, and let $A$ be a symmetric matrix. My understanding is that the Rayleigh quotient of vector $X$ is given by: $$R=\frac{X^T A X}{X^T X}$$ I've been ...
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What is a good technique for testing whether data is Rayleigh distributed?

I have a small data set which (a) is always positive and (b) is showing a right tail on the histogram. I wondered if it could be log-normal and tested for this, but to no avail. I am now wondering ...
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Estimation of parameters in case of a rayleigh random variable corrupted with Gaussian Noise

I have the following model. $$ z(k) = a(k) e^{i \psi(k)} + n(k)$$ The distribution of $a$ is known to be a Rayleigh and $\psi$ is known to be uniformly distributed. The noise $n$ is a white Gaussian ...
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How to derive the solution of $F_S(x)=P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)$?

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*...
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What is the distribution of the ratio of two independent variables, each subject to Rayleigh distribution with different standard deviation?

I am trying to find what is the distribution of the ratio of two independent Rayleigh random variables, each of which has different standard deviation.
complexfilter's user avatar
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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 ...
Efficiency's user avatar
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Why is the square root of a number which is drawn from an exponetial distribution, Rayleigh distributed? [duplicate]

If we have a number $X\sim f(x|\lambda)$ where $$f(x|\lambda) = \lambda e^{\lambda x}, x \geq 0$$ then $\sqrt{X} \sim r(y|\sigma)$ where $$r(y|\sigma) = \frac{y}{\sigma^{2}} e^{-y^{2} /(2 \sigma^{2})},...
user27119's user avatar
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Is my calculation for the MLE correct? How do I check whether it's biased?

In this question, I explored the Rayleigh distribution, with PDF $$f_{\sigma}(x) = \dfrac{x}{\sigma^2} e^{-\dfrac{x^2}{2\sigma^2}},$$ where $x \ge 0$. I calculated that the MLE is $\hat{\sigma^2} = \...
The Pointer's user avatar
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How does $Z = \sqrt{\left( \frac{X_1 - 0}{\sigma} \right)^2 + \left( \frac{X_2 - 0}{\sigma} \right)^2}$ imply that $Z$ has a Rayleigh distribution?

I have two i.i.d. $N(0, \sigma^2)$ random variables $X_1$ and $X_2$. Let $Z = \sqrt{X_1^2 + X_2^2}$. I am told that $R$ follows the Rayleigh distribution. The Rayleigh distribution has PDF $$f_{\sigma}...
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I can't decide if this is a rayleigh or poisson distribution

So here is the distribution it are certain events of time spent behind your telephone. I just can't decide which distribution it follows. I have thought about Poisson or Rayleigh.
Folkert's user avatar
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Statistical test for comparing directional data

I have a dataset with number of bats flying out in 8 directions from their home site (i.e) I studied a bat colony with N individuals and I have counted the number of bats (from that colony) that flew ...
Baheerathan M's user avatar
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How to insert a normal distribution into another function?

I am struggling with the following problem. TLDR: I want to merge the uncertainty of the normal distribution into another function. Imagine a certain significant wave height (Hs) of 2 metres in a sea ...
sjoerdvanhoof's user avatar
2 votes
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Confidence Limits for normal vs rayleigh distributed random variables

I have a device to measure a coordinate in one axis, the measurement error is $\Delta x$, which is specified that $3\sigma < thr$, where thr is say 1 nanometer, and the average is not specified. ...
MiB_Coder's user avatar
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Proof that $f(t) P(x; \sigma = 1)$ is equivilant to $P(x; \sigma = f(t))$ where $P(x; \sigma)$ is a PDF with a single mode

Consider some function or lineshape, $f(t)$, whose form is known. Also consider some one-parameter PDF, $P(x;\sigma)$ where $\sigma$ is a shape parameter -- and importantly defines the mode of $P(x;\...
user27119's user avatar
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Combine normal distribution and Rayleigh distribution

I am trying to find the maximum wave for a given time span based on a given measured wave height. In the top image the measured wave height is 2m and indicated with the black line. The probability ...
sjoerdvanhoof's user avatar
3 votes
0 answers
430 views

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 ...
user27119's user avatar
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The log transformation of a Rayleigh distribution, or identifying the following distributions

I have a set of data which is recorded in a $\log_{10}$ scale. To be exact it is $dBV_{pk} = 20\log_{10}(V_{pk})$ units. If I plot the distribution of the logged data as a PDF I get the following: If ...
user27119's user avatar
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Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$

Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $$T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$$ MY ATTEMPT To begin with, let us notice that \begin{...
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5 votes
1 answer
699 views

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 ...
Anthony Krivonos's user avatar
4 votes
1 answer
2k views

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 ...
Joe Ademo's user avatar
1 vote
1 answer
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Relating Means/Stds Between Gaussian and Rayleigh Distributions

Suppose I have something like a targeting problem, where I specify an angular dispersion in the up and down direction with two Gaussian distributions, each having a mean of 0 and a std of 0.3 degrees. ...
EthanT's user avatar
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2 votes
1 answer
2k views

Finding the parameters of a (possibly) Rayleigh distributed data set

An object is tracked in an experiment I ran. The object's velocities over a 100 time steps are recorded. A model for this object says the velocities should be Rayleigh distributed. Question 1: For ...
troisquatre's user avatar
1 vote
1 answer
334 views

Conditional distribution of multivariate Rayleigh distribution

The correlated Rayleigh envelopes using a set of zero-mean complex Gaussian RVs (Random Variables) is given by $$G_{k}=\sigma_{k}(\sqrt{1-\lambda_k^2}X_k+\lambda_kX_0)+i\sigma_{k}(\sqrt{1-\lambda_k^2}...
yuhou CHEN's user avatar
5 votes
1 answer
336 views

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 ...
CBowman's user avatar
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Assess whether the generated data follows the distribution

Using the Inverse function method I managed to draw a sample of 500 random data from a Cumulative distribution function. $f(x)$,$F(x)$ and $F_X^{-1}(u)$ are as follows: $$f_X=\frac{x}{5}exp\left({\...
Patrick 's user avatar
2 votes
2 answers
2k views

Using a Random number Generator to draw samples from a Cumulative Distribution function

I am given a Rayleigh, distribution function:$$f(x)=\frac{1}{5}x\exp\left(\frac{-x^2}{10}\right)$$ with $x>0$ and asked to: Use an appropriate random number generator algorithm to draw 500 samples ...
Patrick 's user avatar
0 votes
2 answers
2k views

Rayleigh Distribution Quartiles

The Rayleigh distribution has PDF f(x) =xe−$\frac{x^2}{2}$, x >0. Let X have the Rayleigh distribution. (a) Find P(1< X < 3).(b) Find the first quartile, median, and third quartile of X. ...
Mike S.'s user avatar
6 votes
3 answers
551 views

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 $...
user137589's user avatar
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Can deviance residuals from a Rayleigh distribution be negative?

I'm fitting a nonlinear dynamic model to some non-normally distributed data using maximum likelihood estimation. My working approach has been to assume my data is gamma distributed and do gamma ...
K Ball's user avatar
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1 vote
1 answer
907 views

How can I transform a Rayleigh distribution to Standard Normal space?

I need to transform a function of Rayleigh distributed variates $G(X)$ to one in standard normal space $G(U)$. The transformation is governed by equal exceedance probabilities in both spaces, such ...
5DollarBurger's user avatar
7 votes
1 answer
2k views

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 ...
Daniel V's user avatar
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3 votes
1 answer
1k views

Estimating R90 (radius of a circle expected to include 90% of impacts)

I want to determine how big a target I can hit with a bow at a certain distance with 90% probability. I place some paper targets at that distance and shoot 20 arrows at them. I have a ruler and a ...
Louis Stange's user avatar
1 vote
0 answers
361 views

Distribution of the sample mean of a Rayleigh random variable

Let $\{X_1,X_2,X_3\ldots,X_n\}$ be a random sample from a Rayleigh distribution with shape parameter $b$. Let $\bar{X}=(1/n)\sum_{i=1}^nX_i$ denote the sample mean. QUESTION 1: What is the ...
user6006085's user avatar
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1 answer
101 views

Any one knows what the distribution with density $\frac{{x{e^{ - \frac{{{x^2}}}{{2t}}}}}}{{\sqrt {2\pi {t^3}} }}$ is? [closed]

As the title, any one knows what the distribution with density $f(x)=\frac{{x{e^{ - \frac{{{x^2}}}{{2t}}}}}}{{\sqrt {2\pi {t^3}} }}$ is?
Tony's user avatar
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3 votes
2 answers
4k views

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) = \...
Dust009's user avatar
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2 votes
0 answers
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Testing if points on a line are over/under dispersed

I have a series of about 1000 points on a chromosome and I want to know if they are clumped, over-dispersed, or neither. The chromosome can be viewed as a 1-dimensional. I've looked over some ...
lef19's user avatar
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3 votes
1 answer
1k views

Plain english explanation of the Rayleigh distribution?

I need to understand the Rayleigh distribution for a homework assignment in computer networks. Unfortunately, I lack the background knowledge in the field of statistics and probability theory to ...
Kristof Tak's user avatar
5 votes
1 answer
1k views

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 ...
feetwet's user avatar
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5 votes
2 answers
5k views

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's user avatar
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12 votes
1 answer
3k views

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 ...
caracal's user avatar
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5 votes
1 answer
168 views

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 ...
feetwet's user avatar
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13 votes
2 answers
400 views

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 ...
feetwet's user avatar
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