Questions tagged [cauchy-distribution]

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

119 questions
Filter by
Sorted by
Tagged with
37 views

• 1,238
539 views

Median of the squared difference from the median of a Cauchy random variable

Motivation One of the classic challenges with Cauchy random variables is that their moments are not finite, and I even recently learned that Cauchy principal values of even moments of Cauchy random ...
• 8,984
250 views

• 2,169
81 views

Number of samples to estimate Cauchy probability distribution?

I wonder how many samples (approximately) are needed to fit the parameters of a Cauchy probability distribution. I'm guessing probably more than with a normal.
• 333
259 views

Central Tendency of Cauchy distribution

How do you measure the central tendency of a cauchy distribution? I'm aware that the mean is not a good measure of the central location for cauchy. Can I use median? I've been searching with the key ...
• 1,041
1 vote
466 views

Empirical versus Theoretical Convergence of Ratio of Normal Distributions

I have observed that if you take the ratio of two normal variables distributed as $N(1, \sigma) / N(1, \sigma)$, empirically this ratio distribution approaches normality as sigma approaches 0 (shown ...
• 135
6k views

Why is the Cauchy Distribution so useful?

Could anyone give me some practical examples of the Cauchy Distribution? What makes it so popular?
578 views

Probability integral transforms - Cauchy distribution of 1/x and X

When revising for exams, I recently came across the following question: Suppose that $X$ is Cauchy distributed, ie has a density function $$f_X(x) = \frac{1}{\pi(1+x^2)}$$ Show that $1/X$ is ...
• 205
258 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
• 79.5k
1k views

Cauchy distribution: R code [closed]

Generate 1000 sets of numbers from the Cauchy distribution. Do this for set size 2, 5, 10 and 20. Compute the median of each set. Find the distribution of the medians, for each set size. How do I ...
• 101
5k views

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 ...
• 4,109
1 vote
276 views

Sum of powers of standard normal random variables

Context: While trying to teach the Central Limit Theorem I thought it would be a good idea to show a case where it breaks down. Question: Consider the sum of increasing powers of standard normal ...
• 773
4k views

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 ...
• 181
1k views

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 ...
• 1,255
223 views

How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?

Consider a family of distributions with PDF (up to a proportionality constant) given by $$p(x)\sim \frac{1}{(1+\alpha x^2)^{1/\alpha}}.$$ How is it called? If it does not have a name, how would you ...
• 105k
1 vote
48 views

• 133