Distribution Definition and 54 Discussions

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution





{\displaystyle f(x;x_{0},\gamma )}
is the distribution of the x-intercept of a ray issuing from





{\displaystyle (x_{0},\gamma )}
with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.
The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see § Explanation of undefined moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no moment generating function.
In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.
It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

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  1. C

    I Prove that the tail of this distribution goes to zero

    Theorem: Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ## Proof from teacher assistant's notes: We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##: Let ## (s_n)_{n=1}^\infty ## be a...
  2. kfulton

    Winning and losing distributions

    I started by trying to write a normal distribution function for losing and not sure if that is possible.
  3. A

    B It works but why? (Matching experimental data to a random equation)

    Hey guys, I've about a week left to submit my final paper for my trade degree in transportation. The paper is about an analysis of potential implementation of an electric car for direct deliveries in my area where I live. In part of it, I try to analyze how many possible trips a car like...
  4. person123

    I Fitting Data to Grafted Distribution

    I have a set of data (representing the strength distribution of samples), and I would like to fit a normal-Weibull grafted distribution. To the left of a specified graft point, the distribution is Weibull, and to the right it's normal. At the graft point, the value and the first derivative are...
  5. tworitdash

    I Can a Gaussian distribution be represented as a sum of Dirac Deltas?

    We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...
  6. M

    I Failure rate for a uniformly distributed variable

    Hi, I have this question: If random variable T is uniformly distributed over [a, b] , what is its failure rate? Please help
  7. G

    Confusion on the distribution of charge

    The charges are q1,q2 & q. P,Q,O1,O2 refer to positions only. This is a conducting sphere with cavities containing charges. I'm interested in knowing how the charge should be distributed in the sphere. I know the charges induced on the charges of the sphere should be equal and opposite to the...
  8. I

    Bussbar supports

    Why it's not a practice to use such busbar spacing insulator arrangement as it's showed in the picture from attachment (one busbar above other)?
  9. M

    A Product of Gaussian and Rayleigh distributions gives what distribution?

    Hello, I'm trying to find out the distribution function (cumulative or density) of the product of two independent random variables respectively following a non-zero-mean Gaussian and a Rayleigh distribution. The math is too intricate for me, I've found in the appendix of [Probability...
  10. CCMarie

    A Multi-variable function depending on the Heaviside function

    How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
  11. dRic2

    Neutron's crow flight distance & 2° moment of a distribution

    Hi, I'm looking for a simple explanation of the meaning of the crow flight distance and why it is defined as the second moment of a probability distribution: $$\bar r^2 = \int r^2 p(r)dr$$ Where ##p(r)## is the probability that a neutron is absorbed in the interval ##dr## near ##r##. And what...
  12. Vital

    B Define if the distribution is normal or not

    Hello. I wonder how to see if the distribution is normal or not based on the numbers in the table. I attach the screenshot with the table, and it is written that "the distribution appears to be bimodal", so it appears to have two peaks. I understand the the question sounds silly, but I truly...
  13. F

    I Total number density of galaxies and problematic expression

    Hello, I am asked to give the formal expression of the total number density of galaxies and explain why is this expression problematic in practice? From what I saw from my research and into my lectures, I have found the follwing relation which gives the number of galaxies ##N## with mass ##(m...
  14. Vital

    I Understanding part of the multinomial formula

    Hello. I am trying to decipher the formula, making sure I understand what exactly is going on in each part of the expression. I will be grateful for your guidance, corrections and help. Below I show the formula and the example for only 3 possible outcomes (in general it would be k) \(p =...
  15. A

    Electric field distribution in the brain

    I have a PhD in Molecular Neuroscience but am working on something that requires knowledge of electric field distribution in the brain. I understand that electric field intensity (V/cm) is E=(V1-V2)/d in a theoretical model (between 2 infinite plates, that are parallel and equidistant through a...
  16. Vital

    I T-distribution, standard error - how to interpret

    Hello. Math is not my strongest part, yet, so I will be very grateful for your help and for giving the answer in sort of "plain" English, meaning that it will be great if you don't go too deep into the rabbit hole using difficult terminology, as these distribution issues are truly difficult for...
  17. R

    Bayesian Probability Distributions

    Hi, I was having some trouble doing some bayesian probability problems and was wondering if I could get any help. I think I was able to get the first two but am confused on the last. If someone could please check my work to make sure I am correct and help me on the last question that would be...
  18. M

    Automotive How to find the uneven weight distribution on 4 wheels

    Hello, I am working on a project where I am trying to find the weight distribution on a 4 wheeled vehicle that is traveling diagonally up an incline. You can effectively think of the vehicle as having a pitch and a yaw angle in the body frame. Using an arbitrary location of the total center of...
  19. PePaPu

    Thermodynamic assembly - Statistical Thermodynamics

    Homework Statement Consider a model thermodynamic assembly in which the allowed (nondegenerate) states have energies 0, ε, 2ε, 3ε.The assembly has four distinguishable (localized) particles and a total energy U = 6ε. Tabulate the nine possible distributions of the four particles among the...
  20. M

    "Immortal" jellyfish habitat range, Turritopsis dohrnii?

    What is the current habitat range of Turritopsis dohrnii or the "immortal" jellyfish? I know that there are multiple species of the Turritopsis genus and that the species dohrnii have recently increased its range by ballast water spread. What is the current range of the exact species dohrnii?
  21. Euler2718

    I Hypergeometric Distribution Calculation in Libreoffice

    Given this libreoffice command: HYPGEOM.DIST(X; NSample; Successes; NPopulation; Cumulative) >X is the number of results achieved in the random sample. >NSample is the size of the random sample. >Successes is the number of possible results in the total population. >NPopulation is the size...
  22. R

    Marginal Probability Distribution

    Homework Statement Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years): f(xy)=xe^(−x(1+y)) 0 <= x <= y 0 otherwise Find the marginal probability density function of X, fX(x). Enter a formula below. Use * for...
  23. S

    I What is the significance of calculating the average value of cos?

    In kinetic theory, the number of molecules hitting a unit area of a surface per unit time with speeds between v and v + dv and angles between \theta and \theta + d \theta is found to be a function of sin(theta) and cos(theta). There will often be a practice problem asking to show that the...
  24. iikii

    Busy barber problem

    Homework Statement A barbershop has two barbers: an experienced owner and an apprentice. The owner cuts hair at the rate of 4 customers/hour, while the apprentice can only do 2 customers/hour. The owner and the apprentice work simultaneously, however any new customer will always go first to the...
  25. iikii

    Long-run proportion in state ′A′

    The question asks: A physical device can be in three states: A,B,C. The device operates as follows (all time units are in hours): The device spends an exponentially distributed amount of time in stateAA (with mean of 12minutes) and then with probability 0.6 goes to state B, and with...
  26. iikii

    Computer Server Down Probability

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  27. RoboNerd

    Determining relative amounts of work done

    Homework Statement Homework Equations Work = - (integral of) (E dot dl) The Attempt at a Solution Hi, I know that the right answer is D zero, but I fail to understand why. I said that the answer was A as I have different charges, and I thought that depending on how I approach each...
  28. H

    Non-uniform stress distrubution

    Hi everyone! I have perhaps a basic question, but I can't dealt with it. I have a rectangular sample 50x150mm of let's say wood. The sample is compressed from the top over the width of 4mm. I know the shortening of the sample at 70 mm from the bottom (from experimental testing) and I know the...
  29. Ryaners

    (Statistics) Blackbody spectrum in terms of wavelength?

    This is a question about transforming a probability distribution, using the blackbody spectrum as an example. Homework Statement An opaque, non-reflective body in thermal equilibrium emits blackbody radiation. The spectrum of this radiation is governed by B(f) = af3 / (ebf−1) , where a and b...
  30. fluidistic

    I Place pieces on a chessboard randomly until 2 are adjacent

    I've written a program that simulates placing pieces randomly on an nxn chessboard, one by one until 2 are adjacent. When 2 are adjacent I stop, and I report the number of pieces it took. I've plotted a histogram of the number of pieces it took vs number of occurences just to see what kind of...
  31. W

    Pressure tensor reduces to scalar pressure for isotropic dis

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  32. A

    Star/delta transformer - single phase fault on secondary

    Homework Statement A three-phase distribution transformer is configured as follows: -primary side = star with solidly grounded neutral -secondary side = delta, grounded via zigzag earthing transformer A single phase-earth fault occurs on the secondary (delta) side of the circuit...
  33. T

    Pure chance question

    I recently had a discussion with someone about Quantum Mechanics. His story was confusing to me but I could detect that he made an error in his thinking which I proceeded to explain : You are trying to reason from the idea that the 'collapse of the wave-function', which precedes the...
  34. W

    Conditional Expectation of Multiple Independent Random Varia

    Homework Statement Given X,Y,Z are 3 N(1,1) random variables, (1) Find E[ XY | Y + Z = 1] Homework Equations The Attempt at a Solution I'm honestly completely lost in statistics... I didn't quite grasp the intuitive aspect of expectation because my professor lives in the numbers side...
  35. C

    Simple Statistics Question

    The assignment was already turned in a while ago, but I am currently reviewing all the past homework and trying to resolve the problems I couldn't understand. The website software gives the correct multiple choice or numerical answer, but not the steps. They gave me a weird answer and I didn't...
  36. C

    Independent Process Probability distribution

    A manufacturer has designed a process to produce pipes that are 10 feet long. The distribution of the pipe length, however, is actually Uniform on the interval 10 feet to 10.57 feet. Assume that the lengths of individual pipes produced by the process are independent. Let X and Y represent the...
  37. genxium

    How does surface capacitive touchscreen work in detail?

    It's pretty easy to find the "working principle" of surface capacitive touchscreen like http://www.elotouch.com/Technologies/SurfaceCapacitive/howitworks.asp [Broken] Roughly speaking, I know that A surface capacitive touchscreen has a conductive layer to which some electrodes are attached...
  38. E

    Utility Conductor Sizing

    Hello, this is my first post and I've been out of school for a few years and now working a job in my field at a distribution utility, but I'm just really rusty or green on a lot of things now that I'm being challenged. Seems like the simplest questions elude me! Anywho, I've been tasked with...
  39. Kior

    The Probability Density of X^2?

    Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? Question: Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the...
  40. P

    Obtaining Differential and Cumulative Distribution Frequency

    hello all.glad to be here! please in the attached document, column 5 and 6 which represents the differential and cumulative frequency distributions were obtained using equations (5) and (6). I'll be mighty grateful if anyone can give me a step by step breakdown of how it was done?? Many thanks...
  41. M

    I am having difficulty with a Gamma Distribution problem

    Homework Statement Automobile accidents occur in the United States over a 72 hour holiday period like events in a Poisson process with parameter lambda=10/hr. V is the time until the 10th accident a) what is expected value of V or E[V] and standard deviation? b) What is the probability that V...
  42. A

    Voltage behavior in UG cables and OH 11 KV distribution lines

    i want to know the behavior of voltage in 11 KV underground and OH distribution line at different load conditions i.e. no load-full load ...
  43. B

    Calculating Pressure Distribution in a Nozzle for Abaqus

    Hey all, So I have a small convergent nozzle that I'm modelling in Abaqus and I'm wondering how I could work out the pressure distribution so I can apply the correct load. The problem is I only know the pressure being fed to it by air compressor. Is there a way to use Bernoulli s or some other...
  44. H

    Finding the bias of a random sample

    Homework Statement In a study to estimate the proportion of residents in a city that support the construction of a new bypass road in the vicinity, a random sample of 2025 residents were polled. Let X denote the number in the sample who supported the proposal. To estimate the true proportion in...
  45. H

    Uniform Distribution Probability

    Homework Statement The wait time (after a scheduled arrival time) in minutes for a train to arrive is Uniformly distributed over the interval [0,12]. You observe the wait time for the next 95 trains to arrive. Assume wait times are independent. Part a) What is the approximate probability (to 2...
  46. S

    Derivation of the Boltzmann distribution (Dr. David Tong)

    Hello! Dr. David Tong, in his statistical physics notes, derives the Boltzmann distribution in the following manner. He considers a system (say A) in contact with a heat reservoir (say R) that is at a temperature T. He then writes that the number of microstates of the combined system (A and R)...
  47. H

    Exponential Distribution Probability

    Homework Statement The life times, Y in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.6 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if...
  48. C

    Probability Conditional Expectation

    Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ. Find E[Y − X|X + Y = 10] 3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail...
  49. D

    Quick question about normal distributions

    Homework Statement You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes...