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.

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  1. docnet

    Poisson process is identical on equal intervals?

    Let ##N_t## be the Poisson point process with the probability of the random variable ##N_t## being equal to ##x## is given by $$\frac{(\lambda t)^xe^{-\lambda t}}{x!}.$$ ##N_t## has stationary and independent increments, so for any ##\alpha\geq 0, t\geq 0,## the distribution of ##X_t =...
  2. flyusx

    Is a Distribution Function a Ratio of Differentials?

    I read on a post here titled 'Understanding Fourier Transform for Wavefunction Representation in K Space' that if one represents the squared-amplitude as a ratio of differentials, the solution is given. Letting the squared-amplitude be ##\phi##. $$\frac{d\phi}{dp}=\frac{d\phi}{dk}\frac{dk}{dp}$$...
  3. ergospherical

    Drawing from a PDF in python

    Some python function f(x) defines an (unnormalised) pdf between x_min and x_max and say we want to draw x randomly from this distribution if we had the CDF F(x) and its inverse F^{-1}(x), we could take values y uniformly in [0,1], and then our random values of x would be x = F^{-1}(y). but...
  4. Elham1990

    News Plot of Parton distribution function

    Hello I plotted the Parton distribution functions in Mathematica. Now I want to compare the graphs drawn with the graphs of other groups(xu and xd). How should I do this?
  5. N

    A Distribution of sum of two circular uniform RVs in the range [0, 2 pi)

    Hello, I would like to know the analytical steps of deriving the distribution of sum of two circular (modulo 2 pi) uniform RVs in the range [0, 2 pi). Any help would be useful Thanks in advance!
  6. hjam24

    I Write probability in terms of shape parameters of beta distribution

    Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$ If we know that A wins, his score is specified by B's score; he...
  7. Heisenberg2001

    Maxwell-Boltzmann Distribution

    1. ##\vec{p}=m\vec{v}## ##H=\frac{\vec{p}^2}{2m}+V=\frac{1}{2}m\vec{v}^2## ##z=\frac{1}{(2\pi \hbar)^3}\int d^3\vec{q}d^3\vec{p}e^{-\beta H(\vec{p},\vec{q})}## ##z=\frac{Vm^3}{(2\pi \hbar)^3}\int d^3 \vec{v}e^{-\beta \frac{mv^2}{2}}## ##z=\frac{Vm^3}{(2\pi \frac{h}{2\pi})^3}\int d^3...
  8. O

    I Is there a Boltzmann distribution for a system with continuous energy?

    Hi. I'm not sure where to put this question, thermodynamics or the quantum physics forum (or somewhere else). For a system in equillibrium with a heat bath at temperature T, the Boltzman distribution can be used. We have the probability of finding the system in state n is given by ##p_n =...
  9. I

    I Radiant Intensity from Radiant Power and Intensity Distribution

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  10. Z

    I Free particle probability distribution

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  11. A

    MCNP FMESH for Plotting power distribution

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  12. K

    I Geometric Distribution Problem Clarification

    (Geometric). The probability of being seriously injured in a car crash in an unspecified location is about .1% per hour. A driver is required to traverse this area for 1200 hours in the course of a year. What is the probability that the driver will be seriously injured during the course of the...
  13. C

    Exploring Periodic Distribution of Rigid Balls in a Vast Space

    First of all, all the physical quantities presented in this topic are unknown variables, and I require a functional relationship between these unknown variables. In a vast space that does not consider gravity , there are many ideal rigid balls moving freely. And in equilibrium. The ball is...
  14. shaikh22ammar

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    Theorem 11.2.8 in Casella & Berger defines the ANOVA statistic as a maxima of T^2 statistic as: \sup_{\sum a_i = 0} T_a^2 = \sup_{\sum a_i = 0} \left( \left( S^2_p \sum a_i^2 / n_i \right)^{-1/2} \left( \sum a_i \bar Y_{i \cdot} - \sum a_i \theta_i\right) \right)^2 = \left( S^2_p...
  15. rogdal

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  16. M

    I Grating Resolving Power of Laser Beams with Gaussian Distribution

    All resources I’ve found for grating resolving power assume uniform distribution on the grating and produce airy disks. Resolvance is determined by the Rayleigh criterion where the peak of one wavelength is at the minima of the adjacent one. This definition doesn’t seem applicable for Gaussian...
  17. F

    B The historical war of currents in Mains Power Distribution: AC vs DC

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  18. Frabjous

    I Local dark matter distribution

    We do not seem to have any unexplained orbital/gravitational anomalies within the solar system. What does that imply for the local dark matter distribution?
  19. J

    Derive an expression for the radial charge distribution of an E field

    I know we're supposed to attempt a solution but I'm honestly super confused here. I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions. That leaves us with ##\nabla \cdot...
  20. sol47739

    B Shape & Dimensions of Containers: Impact on the Maxwell Boltzmann Distribution

    1.Does the Maxwell Boltzmann distribution change depending on the shape of the container? Pressure and the volume is constant. How is the Distribution affected whether the gas is in: a,sphere b,cube c,cuboid? Why does/doesn’t the distribution change depending on the shape of the container...
  21. B

    Step down transformers used in residential AC Mains distribution

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  22. J

    B Sigma Multiplied Gaussian Distribution

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  23. S

    I Is the Boltzmann energy distribution an instance of energy diffusion?

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  24. S

    B Uniform charge distribution in a conductor

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  25. C

    I Prove that the tail of this distribution goes to zero

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  26. chwala

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  27. chwala

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  28. J

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  29. A

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  30. A

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  31. cosmologyscience

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    Has anyone looked into the details of stellar orbital speeds and required (visible) mass distribution in the Milky Way? Doing some math here - if the local mass density is significantly higher in the inner 10-15% of the galaxy, and then lower and gradually thinning outwards in the disk, we will...
  32. A

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  33. SchroedingersLion

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  34. A

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  35. Zuzana

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  36. Rikudo

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  37. Ranku

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    If two sets of objects, of similar size but different mass, were to be part of a rotating celestial system, how differently would they be distributed? Would the distribution of the lower-mass objects be more spread out, while the higher-mass objects would be concentrated toward the centre?
  38. Ranku

    I Celestial Systems: Mass & Distribution

    If two sets of objects, of similar size but different mass, were to be part of a rotating celestial system, how differently would they be distributed? Would the lower-mass objects be more diffusedly distributed, while the higher-mass objects be more concentrated toward the centre?
  39. A

    A How to derive the sampling distribution of some statistics

    Assume that ##T## has an Erlang distribution: $$\displaystyle f \left(t \, | \, k \right)=\frac{\lambda ^{k }~t ^{k -1}~e^{-\lambda ~t }}{\left(k -1\right)!}$$ and ##K## has a geometric distribution $$\displaystyle P \left( K=k \right) \, = \, \left( 1-p \right) ^{k-1}p$$ Then the compound...
  40. WMDhamnekar

    MHB Math Help: Understand How to Compute $F_{X_1}(x)$

    Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S) Would any member of Math help board explain me that? Any math help will be accepted.
  41. WMDhamnekar

    A Derivation of P.D.F. from distribution function

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  42. chwala

    Solve the problem that involves Normal distribution

    My interest is on part (c), My take, ##Z=\dfrac{160−200}{60}=−0.666666## ##Pr(−0.66666)=0.3546## ##⇒\dfrac{x_1-200}{60}=1.05## ##x_1=63+200=263## Yes, i am aware that they want the answer to ##5## significant figures...i just wanted to check the alternative method... Appreciate your insight...
  43. S

    I Validity of the Boltzmann Distribution

    I (mechanical engineer) have researched this question but can't get to an answer. My question concerns the validity of the Boltzmann distribution. We start with "particles in a box". These particles (at t-zero) may exhibit a range of energies. We place this box of particles in a heat bath for...
  44. A

    A A discrete version of the normal distribution

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  45. L_ucifer

    Understanding Part (b) of a Charge Distribution Problem

    I understand part (a) of this question, and my answer for that part is: *For r < a* E = (ρ0 * r4) / (6 * ε0 * a3) * For r ≥ a* E = (ρ0 * a3) / (6 * ε0 * r2) Now, for part (b), I understand one solution is, for r < a, find the work done to bring a point charge q from infinity to a and then from...
  46. Steve Zissou

    I Distribution of Sum of Two Weird Random Variables....

    Hi there. Let's say I have the following relationship: x = a + b*z + c*y z is distributed normally y is distributed according to a different distribution, say exponential Is there a way to figure out what is the distribution of x? Thanks!
  47. B

    MHB Sampling Distribution of the Sample Means from an Infinite Population

    1. Individual students’ scores on a national test have a normal distribution with a mean of 18.5 and a standard deviation of 7.8. At a Trade School, 84 students took the test. If the scores at this school have the same distribution as national scores, what is the mean, standard deviation and...
  48. A

    Engineering How to calculate the power distribution through this gearbox?

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  49. Barbequeman

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  50. C

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