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
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x
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{\displaystyle f(x;x_{0},\gamma )}
is the distribution of the x-intercept of a ray issuing from
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x
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{\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.
How electric distributers achieve equal phases load, is this done by itself?
Unequal situation hapend only in failure ?
That depend on load, not source, but they cant control load.. so how?
I have a probability distribution over the interval ##[0, \infty)## given by $$f(x) = \frac{x^2}{2\sqrt{\pi} a^3} \exp\left(- \frac{x^2}{4a^2} \right)$$From this I want to derive a formula for the inverse cumulative density function, ##F^{-1}##. The cumulative density function is a slightly...
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 =...
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##.
$$\frac{d\phi}{dp}=\frac{d\phi}{dk}\frac{dk}{dp}$$...
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...
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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?
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!
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...
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 =...
Consider the infrared led TSAL7600 which has the following properties:
$$ \Phi = 35 mW $$
$$ I_e = 25 mW/sr $$
The half angle is ## 30^\circ ## and:
$$ I_r(\theta) = cos^{4.818}(\theta) $$
is a good approximation for the relative radiant intensity.
However, finding the actual radiant...
Abstract:
If a laser shoots photons at a pinhole with a screen behind it, we get a circular non-interference pattern on the screen.
Is this distribution Guassian, and if not, what would its wave function be?
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Assume a double-slit like experiment, but instead of double...
Hello
I'm trying to use FMESH command to get power distribution of this core geometry.
I want to use xyz coordinate in a 1/12 slice of a core so I could use the output of the MCNP sim for a CFD input
How should I approach this?
Thank you
(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...
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...
I have considered two scenarios:
1) A particle that has just collided with the wall at ##z=L## is moving with a velocity ##v_z<0## moving away from the wall. Hence, the probability that this particle has of colliding again is ##0##, so its distribution is also ##0##.
2) A particle moving with...
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...
So, if I get it right, the basic argument goes like this: AC was preferred to DC because its voltage can be stepped up by a transformer. This limits losses while the current is transported from the production plant to the final user. The voltage is subsequently stepped down when delivered to the...
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?
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...
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...
Hello,
If we have a pole top mounted distribution step down transformer for residential use what happens to the voltage on the secondary side if the primary side voltage increase?
Thank you
Hi!
Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously? Is there an equation that easily determines that? Also for other combinations like...
I (mechanical engineer) have researched this question but can't get to an answer.
The equilibrium condition for confined particle diffusion of a solute in a solvent is reached when the solute spatial density is uniform (= zero density gradient), and entropy is max.
But per Boltzmann, when...
How and why can charge be evenly or uniformly distributed in a conductor? How can such near perfect configuration of charge be achieved? Is outside influence (or force) or any special scientific tools or instruments required to accomplish that? By definition, electrostatic equilibrium is...
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...
Consider the attachment below;
How did they arrive at
##F_X (u) = \dfrac{u-a}{b-a}## ?
I think there is a mistake on the inequality, probably its supposed to be ##a≤u<b## and that will mean;
$$F_X (u) =\dfrac{1}{b-a} \int_a^u du= \dfrac{1}{b-a} ⋅(u-a)$$ as required. Your thoughts...then i...
Attached is my reference on the literature.
My question is; ' are there cases where we may have a continuous distribution that has no Mode value? or is it that the Mode will always be there due to the reason that any given function will have a maximum at some point. Cheers.
If a bosonic field is probabalistic, and if it can be emitted (suddenly coming into existence), what determines its probability distribution when it is emitted from a fermion? In other words, one thinks (or at least I think) of a fermion field as already being in existence and already having...
The following is given:
$$\displaystyle P(K = k) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(k -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$
How can you prove that the following equalities are correct?
$$\displaystyle \sum _{k=-\infty }^{\infty }1/2\,{\frac {...
In the article A Discrete Normal Distribution of Dilip Roy in the journal COMMUNICATION IN STATISTICS Theory and methods Vol. 32, no. 10, pp. 1871-1883, 2003 one can read:
A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the
discrete concentration of the normal variate ##X##...
Has anyone looked into the details of stellar orbital speeds and required (visible) mass distribution in the Milky Way?
Doing some math here - if the local mass density is significantly higher in the inner 10-15% of the galaxy, and then lower and gradually thinning outwards in the disk, we will...
I have a probability distribution of the following form:
$$\displaystyle f \left(t \right) \, = \, \frac{\lambda ~e^{-\frac{\lambda ~t }{k }}}{k }, \, 0 < t, \, 0< \lambda, \, k = 1, 2, 3, \dots$$
It seems that this distribution is a limiting case of another distribution. The question is what...
Greetings!
I believe I found an error in a paper to Bayesian neural networks. I think the expression of the covariance of the posterior predictive is wrong, and I wrote down my own calculation. Would be great if a seasoned Bayesian could take a look.
Imagine a regression scenario. We want to...
The shape of the sampling distribution of the Pearson product moment correlation coefficient depends on the size of the sample. Is the expectation of the sampling distribution of the Pearson product moment correlation coefficient always equal to the population correlation coefficient, regardless...
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I try to fit some parameters of the particle (e.g. energy, direction) be means of log-likelihood minimization.
Input data to likelihood function are pulses amplitudes, while Poisson distribution is used. However, the problem is that Poisson distribution is as follows
i.e. for higher...
It is crystal clear that we need torque equation to solve this. But, in order to do so, I need to know where the normal force is located. As far as I'm concerned, normal force is not distributed equally. If this is true, then I suppose this problem is unsolvable? (Though the book says thay it is...
If two sets of objects, of similar size but different mass, were to be part of a rotating celestial system, how differently would they be distributed? Would the distribution of the lower-mass objects be more spread out, while the higher-mass objects would be concentrated toward the centre?
If two sets of objects, of similar size but different mass, were to be part of a rotating celestial system, how differently would they be distributed? Would the lower-mass objects be more diffusedly distributed, while the higher-mass objects be more concentrated toward the centre?
Assume that ##T## has an Erlang distribution:
$$\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$$
Then the compound...
Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S)
Would any member of Math help board explain me that? Any math help will be accepted.
My interest is on part (c),
My take,
##Z=\dfrac{160−200}{60}=−0.666666##
##Pr(−0.66666)=0.3546##
##⇒\dfrac{x_1-200}{60}=1.05##
##x_1=63+200=263##
Yes, i am aware that they want the answer to ##5## significant figures...i just wanted to check the alternative method...
Appreciate your insight...
I (mechanical engineer) have researched this question but can't get to an answer. My question concerns the validity of the Boltzmann distribution.
We start with "particles in a box". These particles (at t-zero) may exhibit a range of energies. We place this box of particles in a heat bath for...
I have the following function for the normal distribution:
$$\displaystyle f \left(x \right) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$
How can the following integrals be equal to their sums?
$$\displaystyle \int_{-\infty...
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...
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!
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...
Greeting!
My problem is to understand the flux of power inside this gearbox
for example I can´t understand why the power in load 2 and load 1 are not equal to the power of the motor (suppose 100% efficiency). if this gearbox was an energy circuit (how to know that the energy is transmitted and...