What is Distribution: Definition and 1000 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



f
(
x
;

x

0


,
γ
)


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



(

x

0


,
γ
)


{\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.

View More On Wikipedia.org
Recent contents Top users
  1. Barbequeman

    Consider three models of the Mass distribution in the Milky Way

    I attached a Jpeg with my attempted solution but I´m not sure if I´m on the right way... I hope for a correction in my calculations
  2. C

    A Perturbation of Maxwell stress from voltage distribution

    I have a voltage distribution ##V(x,y) = V_{dc}(x,y)+ V_{ac}(x,y) \cos(\omega t)##, I have derived the Matrix e. But I do not know how to extract it from the voltage, meaning I do not know how to find ##E_{x0} , E_{y0}, \delta E_{x}, \delta E_{y}## in terms of ##V_{dc}(x,y), V_{ac}(x,y)##...
  3. barryj

    Is My Understanding of Poisson Distribution Correct for Different Area Sizes?

    Please excuse me for posting in this group.l There seems to be little activity in the statistics group and i might get no response. For this example, lambda, is the average of dogs per 100 square miles = 0.05. if I wanted the probability of 2 dogs in 100 square miles I would calculate.. P(x=2)...
  4. G

    Force on a particle of a linear charge distribution

    Hello! I am trying to solve this exercise of the electric field, but it comes out changed sign and I don't know why. Statement: On a straight line of length ##L=60\, \textrm{cm}## a charge ##Q=3,0\, \mu \textrm{C}## is uniformly distributed. Calculate the force this linear distribution makes...
  5. L

    Finding the distribution of random variables

    Hi. I have found the answer to a and c (I don't know whether it is correct) but I do not know what I should find in question b. Is my method correct and how should I solve part b? Thank you for your help!
  6. J

    Question about Pylons (AC Mains Distribution Towers)

    Hi, Please could someone explain in detail the arrangements of transmission cables in a UK pylon? In the image below there seems to be 3 sets of two wires on each side. As I understand it UK power stations produce 3-phase electricity so does each side come from two power stations? So this pylon...
  7. M

    I Finance: does the volatility of a stock follow a certain distribution?

    Hi, I am not sure whether this is the right forum to post this. Please let me know if I should move and will do so. Overall question: Does the 'volatility' (i.e. standard deviation of the log returns) follow any sort of statistical distribution - maybe normal or log-normal? Background/...
  8. Daniel777

    Investigating the Charge Distribution on a Bulging Sphere

    In this question it is given that the sphere which is conducting is initially given a charge q then due to nonuniform mechanical strength and due to electrostatic force it creates a Small hemispherical bulge on its surface? okay my doubt is Let me define a term σ where σ is surface density...
  9. warhammer

    Intensity Distribution of Superposition of 2 Waves

    We assume incident waves to be: y(1)=y(o)sin(wt) y(2)=3y(o)sin(wt+Φ) As Intensity~(Amplitude)^2 We get y(2)=3y(1) This gives us I(2)=9I(1) We assume I(1)=I(o) & I(2)=9I(o) Resultant Wave Intensity I=I(1)+I(2) +2√(I(1)*I(2))*cosΦ ----> I(o) + 9I(o) + 6I(o)cosΦ (We can take cos of this...
  10. Leo Liu

    I Gauss' law and an object with nonuniform charge distribution

    Gauss' law: $$\iint_{\partial A}\vec E\cdot d\vec A=\frac{Q}{\epsilon_0}$$ Suppose we have a unevenly charged non-conducting spherical shell, in which a Gaussian surface is placed. In this case, is the electrical field on A 0, given that there is no charge inside A? I came up with this example...
  11. riay00

    I How to create a fair distribution algorithm?

    Hi, I am a math noob and need to understand what is the best mathematical formula for fair distribution. The problem statement is as follows: We choose 2 numbers at a given time to distribute donations at a particular instance in time (The numbers from 1-6500 represent community members...
  12. F

    B Field distribution in semiconductors

    Hello for everyone. I have a question according the field distribution in the semiconductor while the field effect. According to logic, the field is scrreened due to the field of the polarized carriers like electrons and holes. I know about the Debiye length. And that the field on the infinity...
  13. J

    Calculate lift from pressure distribution

    v=50m/s ∆Pg= -981Pa ∆𝑝𝑑 =490Pa c=1m 𝜌 = 1,225 kg m3 𝐹3 = ∫ ∆𝑝3 ∙ 1 ∙ 𝑑𝑥 𝑐 2 0 = ∆𝑝𝑔 ∫ 2∆𝑝𝑑 𝑐 𝑥𝑑𝑥 𝑐 2 0 = 2∆𝑝𝑑 𝑐 ∙ ( 𝑥 2 2 ) c⁄2 | 0 = 2∆𝑝𝑑 𝑐 ∙ 𝑐 2 4 ∙ 2 = ∆𝑝𝑑 ∙ 𝑐 4 = 490,5 ∙ 1 4 = 122,625 N Lift per unit of span = (F1) + (F2) +F3 + F4= 490.5+245.25+122,25+122,625=981N :confused: Pressure...
  14. ASKINGHUMAN

    Confused about calculating the average energy from a distribution graph

    Hello, in one of tasks of my liquid scintillation lab is to determine the average energy. You can see from the graph that data I obtained is similar to this one that I have a excel sheet data. X-axis is for beta particle energy from 0-156keV while y-axis counts of the particles. So according to...
  15. evceteri

    Neutron scattering probability distribution

    Hi, I'm reading Chapter 2-II of of Duderstadt & Hamilton's "Nuclear Reactor analysis". In the section "Differential scattering cross sections with upscattering" it is discussed the situation in which neutrons suffers elastic scattering collisions in a hydrogen gas at finite temperature T and the...
  16. kfulton

    Winning and losing distributions

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

    MHB How to prove the normal distribution tail inequality for large x ?

    What is the meaning of this proof? What is the meaning of last statement of this proof? How to prove lemma (7.1)? or How to answer problem 1 given below?
  18. chwala

    Solve the probability distribution and expectation problem

    This is the problem; Find my working to solution below; find mark scheme solution below; I seek any other approach ( shorter way of doing it) will be appreciated...
  19. 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...
  20. chikchok

    I Fermi energy definition and Fermi-Dirac distribution

    1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent...
  21. chikchok

    I Is the Fermi-Dirac distribution equal to zero at the state of highest energy?

    I`m sorry if this seems too obvious, just trying to clarify something. When Fermi-Dirac distribution is equal to zero , can we assume it is the state of the highest energy? (Because the propability of occupation is zero)
  22. mcas

    Fermi-Dirac distribution at T->0 and \mu->\epsilon_0

    The limit itself is pretty easy to calculate ##lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}## But I'm very confused about changing ##\epsilon_\vec{k}## to ##\epsilon_F##. Why do we do this?
  23. R

    I Boltzmann Distribution: Formula & Fig 2a in Document

    Hello Can anyone explain what formula (or parameters) was used to create the exponential Boltzmann distribution in fig 2a of this document? http://image.sciencenet.cn/olddata/kexue.com.cn/upload/blog/file/2009/5/20095251352697121.pdf I figure it must be something like y=e^(ln(600)-b*x) for some b?
  24. M

    A Uniform distribution setup

    A random variable is distributed uniformly over a circle of radius R. What does the cdf ##F(x,y)## look like as a function of the Cartesian coordinates? The pdf can be expressed as ##f(x,y)=\frac{\delta(\sqrt{x^2+y^2}-R)}{2\pi R}##, where ##\delta## is Dirac delta function. Integration is...
  25. V

    Uneven charge distribution on a conductor

    All I can say is that where the charge density on surface is higher, we will have a stronger electric field compared to areas where charge density is lower since more charges means greater electrical force on a test charge placed very close to the surface. Also, the potential on pointed areas...
  26. PainterGuy

    I Distribution function and random variable

    Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg
  27. V

    Volume density vs Surface density of charge distribution

    This doubt is confusing to me. I know it's something to do with conductors and insulators, but cannot explain. Conductors have mobile/free electrons unlike insulators. Having free electrons doesn't seem to explain this difference of charge distributions.
  28. B

    Expressions for energy & entropy from free energy (discrete distribution)

    So just by by using the definition of the partition function... $$ Z = \sum_i e^{ \frac {-E_i} {k_BT} } = e^{ \frac {-0} {k_BT} } + e^{ \frac {-\epsilon} {k_BT} } = 1 + e^{ \frac {-\epsilon} {k_BT} } $$ And then, a result we obtained in class by using the Boltzmann H factor to solve for ##S##...
  29. M

    Intersection of two line segments from uniform distribution

    Hi, I found this question online and made an attempt and would be keen to see whether I am thinking about it in the right manner? Question: Find the probability of two line segment intersecting with each other. The end points of lines are informally sampled from an uniform distribution...
  30. M

    Bayes Theorem: coin flips and posterior predictive distribution

    Hi, I was attempting the following question and just wanted to check whether my working was correct: Question: A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the...
  31. hagopbul

    I Another question from Ashcroft and Mermin: Fermi-Dirac Distribution

    Good Day : i reached the page 40 of Ashcroft Mermin book and after the equation 2.38 there is this expression of E(a,N) which is equal to Helmoltez Free energy F = U - TS , how this two terms F , E are related ? anyone can provide adequate explanation , and few useful references Best...
  32. karush

    MHB Normal distribution graph P(a<x<b)

    The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram.\\ Write down the values of a and b. $a=\dfrac{2}{4.4}= 0.455 b=\dfrac{3}{4.4}=0.682]$ ok this was a key to a test question from 2013 but mostly...
  33. W

    What Causes the Non-Circular Distribution of Magnetic Fields in Steel Ducts?

    Hi all, first time on this forum. I know this may sound like a stupid question, but how does the magnetic field distribute? I am working on FEMM and i am analysing magnetic losses on steel ducts. I was checking the flux density and the magnetic field distributions and i was surprised when i...
  34. Lynch101

    I The Probability Distribution and 'Elements of Reality'

    The below comment by @vanhees71 is an interesting one and I would be interested in exploring its implications. I am inclined to think that we can draw certain inferences about nature based on how we interpret the probability function and what it tells us about the elements of reality of the...
  35. Ackbach

    MHB Uniformly Most Powerful Test for Weibull Distribution

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $Y_1,\dots,Y_n$ be a random sample from the probability density function given by $$f(y|\theta)= \begin{cases} \dfrac1\theta\,m\,y^{m-1}\,e^{-y^m/\theta},&y>0\\ 0,&\text{elsewhere}...
  36. Ackbach

    MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...
  37. Ackbach

    MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...
  38. warhammer

    Maxwell's Distribution Law (Thermal Physics)

    There are two questions in the photo. I have attempted the solution (attached below) and I would be highly obliged if someone would verify the same. Edit- Sorry the images of the solution have uploaded in the wrong order. 5th and 1st Image comprise of both parts of Q1 while the remaining of Q2.
  39. Leo Liu

    Equation connecting potential and potential energy of a distribution

    The equation below allows us to calculate the potential energy of a continuous distribution of electric charge. $$U=\frac {\epsilon_0} 2 \iiint\limits_\text{Entire electric field}\vec E^2\,dV$$ In my textbook, the author states $$U=\frac 1 {8\pi\epsilon_0}\iiint\limits_\text{Entire electric...
  40. Phys pilot

    I Energy distribution plot of neutrinos in beta decay

    Hello, When you have a beta decay you get the typical continuos spectrum representing counts against the kinetic energy of the electron. But what's the shape and how I get the spectrum of the kinetic energy of the neutrinos? Thanks
  41. 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...
  42. merlyn

    B Maxwell-Boltzmann distribution under the influence of a gravitational field

    Could anyone suggest a simple video showing the Maxwell-Boltzmann distribution under the influence of a gravitational field? I trying to show a flat earther idiot how pressure gradients arise in a simple manner. Thank you all. DF
  43. 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) =...
  44. H

    B Distribution of energy in the electric field surrounding an electron

    I am thinking about how an electric field has energy associated with it. If a single electron exists alone in a remote vaccuum, I believe it has it's own electric field surrounding it, and that this field has an energy content associated with it. My question is; does this electric field store...
  45. Omega0

    B Distribution of particles, given a function

    I am not that super expert of statistics, so feel free to shift my formulation of the problem into the right one.First, for a physicist, the basic formulation of the problem. Let us say that you have a gravitational field and you have a fully symmetric problem on a flat world without other...
  46. Viky1147

    A Granular convection (particle distribution) calculation

    Hi All, Am new here came particularly hoping for some guidance I need to calculate theoretically, how different types of particles (shown below) distribute in a granular convection Condition: Large surface or open surface Types of particles: 1) Low volume, Low density particle 2) High...
  47. F

    A Analytical expression of Cosmic Variance - Poisson distribution?

    I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
  48. S

    Probability related to Normal Distribution

    a) Let X = distance walked on Friday and Y = distance walked on Saturday X ~ N (12, 0.192) and Y ~ N (10, 0.52) Let A = Y - X → A ~ N (-2 , 0.2861) P(Y > X) = P(Y - X > 0) = P(A > 0) = 9.2 x 10-5 But the answer key is 0.026 Where is my mistake? Thanks
  49. M

    Probability: Prior Distribution for a Continuous Variable

    Hi, I was attempting this problem from the MIT OCW website probability and statistics course. Context: When Jane started class, she warned Jon that she tends to run late. Not just late, but uniformly late. That is, Jane will be late to a random class by ##X## hours, uniformly distributed over...
  50. U

    A question on the Dirac delta distribution

    Is it correct to say that $$\int e^{-i(k+k’)x}\,\mathrm{d}x$$ is proportional to ##\delta(k+k’)##?
Back
Top