Skip to main content

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
37 views

Ratio of Normal Distributions [duplicate]

Suppose I have two independent random variables, $X \sim N(\mu_1,\sigma_1^2)$ and $Y \sim N(\mu_2,\sigma_2^2)$ with $\mu_1,\mu_2 > 0$. How can I compute/estimate $$ \mathbb{E}\left[\left\lvert \...
Algebro1000's user avatar
5 votes
3 answers
275 views

Divergence of $e_{i+1}\leftarrow e_i - x_i e_i$ for Cauchy $x_i$

Suppose $e_0=1$ and $e_k$ evolves according to the following recurrence with $x_i\sim \operatorname{Cauchy}$, IID draws from standard Cauchy random variable. $$e_{i+1}\leftarrow e_i - a (x_i e_i)$$ ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
36 views

Why does the t-SNE paper claim that "large clusters of points that are far apart interact in just the same way as individual points"?

In the original t-SNE paper, the authors explain the use of the t-distribution with one degree of freedom (i.e. Cauchy distribution) for the map points, $(1 + |y_i - y_j|^2)^{-1}$, as follows: ...[it]...
Denziloe's user avatar
  • 1,133
6 votes
0 answers
222 views

Running maximum of $\sum_{1\leq k\leq n} X_i$ for Cauchy random variables $X_i$

Suppose $X_i$ are $\mathrm{Cauchy}(0,~\gamma)$ IID RV's and let $S_n=X_1+\cdots+X_n$ be their sum. Does an expression exist for the CDF of the running maximum up to an index $1 \leq k \leq n$? Edit: ...
user169291's user avatar
0 votes
0 answers
29 views

Comparison between X/Y and X/|Y| when X and Y are iid Normal(0,1) (Cauchy distribution) [duplicate]

I was studying mathematical statistics written by Joe Blitzstein when I encountered the derivation of the PDF of Cauchy distribution. I cannot understand why the distribution is the same between X/Y ...
Robin311's user avatar
  • 155
0 votes
0 answers
27 views

Can all MDPs be translated into MPs?

Say we have a Markov Decison Process (MDP) with countably infinite state space $S$, continuous action space $A$, well-defined transition matrix $T_a(s,s') = P(s'\mid s,a)$ and some stochastic policy $\...
Joonas's user avatar
  • 1
3 votes
1 answer
265 views

Let $X_1,X_2,\ldots$ be iid random variables with Cauchy distribution and $S_n=X_1+X_2+\cdots+X_n$, find $P(S_n>an)$, $a>0$

Let $X_1,X_2,....$ be iid random variables with Cauchy distribution and $S_n=X_1+X_2+\cdots+X_n$, find $P(S_n>an)$, $a>0$. This is exercise 8.44 of the intro book of Grimmet and Welsh. We cannot ...
muhammed gunes's user avatar
11 votes
2 answers
519 views

How much would you wager for a Cauchy distributed return?

Suppose there's a casino that has a game where you sample from a Cauchy (100, 1) distribution (mode is 100). If the sample is positive, then the casino pays you that amount, otherwise you'd have to ...
George Chang's user avatar
1 vote
0 answers
95 views

OLS with $iid$ Cauchy errors still unbiased?

A comment to this question suggests that the OLS estimate of linear model parameters is unbiased, even when the error term is Cauchy. Given that Cauchy distributions lack an expected value, I am ...
Dave's user avatar
  • 63.7k
2 votes
2 answers
490 views

Linear regression with Cauchy distribution for errors

I have ran the below linear regression model and using the performance package in R I however checked whether the distribution of the residuals is normal. The performance package suggests I should be ...
luciano's user avatar
  • 14.4k
2 votes
1 answer
139 views

CDF of max of $n$ cauchy variates

Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf $$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$ Define: $$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
AgnostMystic's user avatar
0 votes
1 answer
39 views

"Slack" variance and the Cauchy distribution

I was staring at a time series and thought an interesting way to measure a variance-like value would be to treat the time series like a physical chain and the amount of slack in that chain would be ...
Jemmy's user avatar
  • 132
4 votes
2 answers
324 views

Why does my probability plot not agree with my histogram?

I am trying to determine the best distribution to characterize a dataset. My dataset visually looks like Cauchy distribution would be a good fit, so I used python to fit a Cauchy distribution to my ...
Andrew's user avatar
  • 41
1 vote
0 answers
96 views

Arithmetic with normally-distributed variables

Suppose $x \sim \mathcal N(\mu_x,\sigma_x^2)$ and $y \sim \mathcal N(\mu_y,\sigma_y^2)$ are random variables, and suppose $\mu_y$ is large compared to $\sigma_y$. I want to know about $$ z=\frac{x}{y^...
Charles's user avatar
  • 1,238
9 votes
1 answer
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 ...
Galen's user avatar
  • 8,984
7 votes
1 answer
250 views

Do the principal values of higher moments exist for the Cauchy distribution?

It is known that the Cauchy distribution has undefined moments, and that the expectation has a principal Cauchy value $\operatorname{PV}\left( \mathbb{E} [X] \right)$ of zero. I wonder if $\...
Galen's user avatar
  • 8,984
0 votes
0 answers
14 views

Kullback Leibler divergence between normal and Cauchy distributions? [duplicate]

Let $\phi(x)$ be the standard normal probability density function and $f(x)$ be the Cauchy probability density function. How can I calculate the Kullback-Leibler divergences between $\phi$ and $f$. ...
Cochon's user avatar
  • 1
14 votes
3 answers
976 views

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. ...
Greenparker's user avatar
  • 15.7k
4 votes
3 answers
884 views

Maximum likelihood estimator of Cauchy distribution but with a catch

I have an exercise to solve that states that we need to find the Maximum likelihood estimator of location parameter of the Cauchy distribution given a set X={x1,x2} , |x1-x2|<2 Now I worked using ...
Giannhs Meh's user avatar
17 votes
1 answer
966 views

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 ...
Simon Segert's user avatar
  • 2,034
2 votes
0 answers
59 views

Need help understanding the geometric reasoning that a Cauchy distribution has no mean

I need help understanding the reasoning that the Cauchy distribution can't have a finite mean in this answer: https://stats.stackexchange.com/a/36131/25186. It uses some geometric intuition arising ...
ryu576's user avatar
  • 2,540
1 vote
1 answer
78 views

Can we predict what happens to the sample mean as we increase sample size if the true mean blows up?

The Cauchy distribution is used as an example of a pathological case where the mean blows up. For such a distribution, we can imagine drawing samples and tracking the sample mean as the number of ...
ryu576's user avatar
  • 2,540
3 votes
2 answers
192 views

What happens when you try to find standard deviation of a (non-truncated) cauchy distribution? [duplicate]

I have read that this doesn't work, but I do not understand exactly why. Please can someone explain.
milkcookie's user avatar
2 votes
1 answer
262 views

Conditional distribution of multivariate cauchy distribution

In the example of multivariate normal distribution, $$ \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{...
user331385's user avatar
1 vote
1 answer
400 views

Why do I get two different mean values with two different methods for the same sample?

I have this dataframe where I created the 3rd column using the first two columns. Both ${Y}$ and ${X}$ are independent random variables. $\bar{Y}$ $\bar{X}$ $\bar{Z} = 100\frac{\bar{X}-\bar{Y}}{\bar{...
gülsemin's user avatar
0 votes
1 answer
98 views

Proving the average of sum of i.i.d cauchy is not a consistent estimator of location parameter

Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$ To prove that it is inconsistent, consider the characteristic function of $X_i$ and $\bar{X}$, which ...
smaillis's user avatar
  • 123
1 vote
1 answer
1k views

Why is Cauchy the default prior for both testing and estimation?

Assume that a data set follows a normal distribution and the prior and posterior both have a normal-gamma distribution. When we are performing Bayesian analysis but don't want any subjective choice of ...
Lerner Zhang's user avatar
  • 6,748
0 votes
1 answer
62 views

Distributions other than $N(0, 1)$ for the probit-style regression link function

When we do a probit regression, we use the distribution of a standard normal to convert from the linear combination of the predictors to a probability value. Why stop at the standard normal? Why not ...
Dave's user avatar
  • 63.7k
-1 votes
1 answer
282 views

Normalize the posterior density for a Cauchy Distribution C ($\theta$,1) and a Uniform [0,100] prior [closed]

Using a Bayesian approach we have $$P(\theta|\text{data})= P(\text{data}|\theta) \frac{P(\theta)}{P(\text{data})}$$ Therefore, the posterior distribution will be proportional to $$\frac{1}{N} (1+(y+\...
Roshni Modi's user avatar
0 votes
1 answer
105 views

Facing difficulties in Cauchy PDF problem from harvard stats 110 book [duplicate]

I have two doubts: If X and Y are independent then PDF of X/Y is simply PDF of X(by independence) I don't know where am I interpreting wrong, Please correct me. How X/Y and X/|Y| are identically ...
Gulshan Arya's user avatar
0 votes
1 answer
198 views

Cauchy Distribution in R [closed]

Cauchy distribution. Draw 1000 sets of numbers from the Cauchy distribution using set.seed(100). Do this for set size 2, 5, 10 and 20. Compute the median of each ...
user avatar
1 vote
3 answers
2k views

Sampling from heavy vs light tailed distribution

I am having some issue understanding the behavior of such distributions when generating random numbers. I was under the impression that heavy tailed distributions have "heavier" tails, so ...
Marco De Virgilis's user avatar
8 votes
2 answers
658 views

Does there exist a Frequentist or Non-Bayesian solution to Gull's Lighthouse Problem?

Does there exist a Frequentist or ODE or Non-Bayesian solution to Gull's Lighthouse Problem which is correctly modeled with cauchy distribution? See The Lighthouse Problem and Dave Harris' answer to ...
user avatar
1 vote
1 answer
313 views

Bias-Variance Trade Off with Cauchy Estimator

I'm having a look at the bias and standard error of a set of estimators. I expected to see the trade off when varying the parameter of the estimator, but I see that both the bias and the variance ...
JoBel's user avatar
  • 11
3 votes
1 answer
1k views

Ratio of two independent normal : cumulative sum [duplicate]

Given two independent normal variables, $X\sim N(0, \sigma^2_X)$ and $Y\sim N (0, \sigma_Y^2)$, find the probability that $\frac{X}{Y} < 1$. I seem to prove: $Z = \frac X Y \sim $ Cauchy ...
hola's user avatar
  • 131
12 votes
2 answers
5k views

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&...
Ron Snow's user avatar
  • 2,169
0 votes
1 answer
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.
Ambesh's user avatar
  • 333
2 votes
1 answer
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 ...
Eric Kim's user avatar
  • 1,041
1 vote
1 answer
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 ...
Zane Blanton's user avatar
21 votes
2 answers
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?
2 votes
1 answer
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 ...
Jhonny's user avatar
  • 205
5 votes
1 answer
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 ...
kjetil b halvorsen's user avatar
-1 votes
2 answers
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 ...
JKJ's user avatar
  • 101
16 votes
2 answers
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 ...
mavavilj's user avatar
  • 4,109
1 vote
0 answers
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 ...
S. Punky's user avatar
  • 773
8 votes
3 answers
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 ...
Ignacio's user avatar
  • 181
12 votes
2 answers
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 ...
Chechy Levas's user avatar
  • 1,255
8 votes
1 answer
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 ...
amoeba's user avatar
  • 105k
1 vote
0 answers
48 views

Formutaion of Least Squares Problem

In general, to use the method of least squares, a linear stochastic system is modeled as: \begin{equation} y = ax + \eta \end{equation} where, $y$, is an observed variable, $x$ is an input while $\...
user146290's user avatar
3 votes
1 answer
2k views

Find the Maximum Likelihood Estimator given two pdfs

From the book Introduction to Mathematical Statistics by Hogg, McKean and Craig (# 6.1.12): Let $X_1,X_2,\cdots,X_n$ be a random sample from a distribution with one of two pdfs. If $\theta=1$, then $...
evaristegd's user avatar