Cauchy distribution is a symmetric density which equals the t distribution with one degree of freedom. The expectation and variance of the cauchy distribution do not exist. See https://en.wikipedia.org/wiki/Cauchy_distribution
Questions tagged [cauchy-distribution]
107 questions
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votes
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Why does the Cauchy distribution have no mean?
From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
Flying pig
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What are the properties of a half Cauchy distribution?
I am currently working on a problem, where I need to develop a Markov chain Monte Carlo (MCMC) algorithm for a state space model.
To be able to solve the problem, I have been given the following probability of $\tau$: p($\tau$) =…
Christoph
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Cauchy Distribution and Central Limit Theorem
In order for the CLT to hold we need the distribution we wish to approximate to have mean $\mu$ and finite variance $\sigma^2$. Would it be true to say that for the case of the Cauchy distribution, the mean and the variance of which, are undefined,…
JohnK
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Why is the Cauchy Distribution so useful?
Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?
Daria
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What is the distribution of sample means of a Cauchy distribution?
Typically when one takes random sample averages of a distribution (with sample size greater than 30) one obtains a normal distribution centering around the mean value. However, I heard that the Cauchy distribution has no mean value. What…
Molossus Spondee
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Is Cauchy distribution somehow an "unpredictable" distribution?
Is Cauchy distribution somehow an "unpredictable" distribution?
I tried doing
cs <- function(n) {
return(rcauchy(n,0,1))
}
in R for a multitude of n values and noticed that they generate quite unpredictable values occasionally.
Compare that to…
mavavilj
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What do high dimensional cauchy distributions look like?
A well-known rule of thumb is that for high dimensions $d$, the Gaussian distribution $N(0,I_d)$ is approximated by the uniform distribution on a sphere $U_{\sqrt{d}S^{d-1}}$. This has been mentioned on previous cross validated posts, e.g. Why is…
Mike Hawk
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MLE of the location parameter in a Cauchy distribution
After centering, the two measurements x and −x can be assumed
to be independent observations from a Cauchy distribution with
probability density function:
$f(x :\theta) = $ $1\over\pi (1+(x-\theta)^2) $ $, -∞ < x < ∞$
Show that if $x^2≤ 1$ the…
user123965
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Kernel Density Estimate for Cauchy
As far as I understand, kernel density estimation does not make any assumptions on the moments of the underlying density, and just requires smoothness. The Cauchy density function is quite smooth. Even still, when I try to do KDE using density() in…
Greenparker
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Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?
If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread.
Does this property have a name?
Are there any other distributions for which this is…
Chechy Levas
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How can I obtain a Cauchy distribution from two standard normal distributions?
I am interested in
Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$
is a Cauchy random variable.
My work:
$f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty
Jen Snow
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Maximum likelihood estimator of location parameter of Cauchy distribution
I have reached up to
$$\frac{d\ln L}{d\mu}=\sum_{i=1}^n \frac{2(x_i-u)}{1+(x_i-u)^2}$$
Where $u$ is location parameter.
And $L$ is likelihood function.
I'm not getting how to proceed. Please help.
user89929
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Expression for the median of a sum of half-Cauchy random variables
This is a technical problem I encountered in research. Sorry if it reads simple for professional statisticians.
Let $X_1, \cdots, X_n$ be iid standard Cauchy random variables. What estimate or exact formula exists for the median of $\sum_{i=1}^n…
Ju Sun
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Is the sum of a large number of independent Cauchy random variables Normal?
By Central Limit Theorem, the probability density function of the the sum of a large independent random variables tends to a Normal. Therefore can we say that the sum of a large number of independent Cauchy random variables is also Normal?
urvah shabbir
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Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
Given Cauchy distribution with pdf $p(x) = \frac{1}{\pi ((x - \theta)^2 + 1)}$
how can I find a consistent unbiased estimator for $\theta$?
My reasoning so far
Tried MLE, but there seems to be no closed-form expression. See Maximum likelihood…
Ignacio
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