Science topic

# Probability - Science topic

Probability is the study of chance processes or the relative frequency characterizing a chance process.

Questions related to Probability

Pasricha, Satwant K. “Relevance of para-psychology in psychiatric practice.” Indian journal of psychiatry vol. 53,1 (2011): 4-8. doi:10.4103/0019-5545.75544

In my own words, possibly reading minds, possible afterlife and probability.

What’s the most common programming paradigm of no-code platforms? Why?

Good day! The question is really complex since CRISPR do not have any exact sequence - so the question is the probability of generation of 2 repeat units, each of 23-55 bp and having a short palindromic sequence within and maximum mismatch of 20%, interspersed with a spacer sequence that in 0.6-2.5 of repeat size and that doesn't match to left and right flank of the whole sequence, in a random sequence.

Quantum mechanics focuses more on probability and specific units which seems more empirical. Whereas relativity is more theoretical and thus rationalist.

"DNA is SO unpredictable that they are either fractals or something less predictable, thus a gene is never known to manifest into a trait, debunking hereditarianism and vindicating CRT" (Ohnemus 2024).

I get \lambda_i in the numerator when I integrate instead of \lambda_1.

Afterlife: Universalist Christian Heaven

Epistemology: falsifiability and skeptical empiricism

Ethics: deduced from tradition, then risk analysis, and lastly skin in the game. Manifested as natural law(moderation and negative utilitarianism), political correctness.

Politics: progressivism and open society.

Respectfully, diversely, equitably and inclusively, who agrees the lack of moral absolutes(morality is objective but relative) decreases the likelihood of reincarnation? How?

Discussion Stimuli:

**There are many kinds of certainty in the world, but there is only one kind of uncertainty**.

I: We can think of all mathematical arguments as "causal" arguments, where everything behaves deterministically*. Mathematical causality can be divided into two categories**: The first type, structural causality - is determined by static types of relations such as logical, geometrical, algebraic, etc. For example, "∵ A>B, B>C; ∴ A>C"; "∵ radius is R; ∴ perimeter = 2πR"; ∵ x^2=1; ∴ x1=1, x2=√-1; .......The second category, behavioral causality - the process of motion of a system described by differential equations. Such as the wave equation ∂^2/ ∂t^2-a^2Δu=0 ...

II: In the physical world, physics is mathematics, and defined mathematical relationships determine physical causality. Any "physical process" must be a parameter of time and space, which is the essential difference between physical and mathematical causality. Equations such as Coulomb's law F=q1*q2/r^2 cannot be a description of a microscopic interaction process because they do not contain differential terms. Abstracted "forces" are not fundamental quantities describing the interaction. Equations such as the blackbody radiation law and Ohm's law are statistical laws and do not describe microscopic processes.

The objects analyzed by physics, no matter how microscopic†, are definite systems of energy-momentum, are interactions between systems of energy-momentum, and can be analyzed in terms of energy-momentum. The process of maintaining conservation of energy-momentum is equal to the process of maintaining causality.

III: Mathematically a probabilistic event can be any distribution, depending on the mandatory definitions and derivations. However, there can only be one true probabilistic event in physics that exists theoretically, i.e., an equal probability distribution with complete randomness. If unequal probabilities exist, then we need to ask what causes them. This introduces the problem of causality and negates randomness. Bohr said "The probability function obeys an equation of motion as did the co-ordinates in Newtonian mechanics "[1]. So, Weinberg said of the Copenhagen rules, "The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics" [2].

IV: The wave function in quantum mechanics describes a deterministic evolution energy-momentum system [3]. The behavior of the wave function follows the Hamiltonian principle [4] and is strictly an energy-momentum evolution process***. However, the Copenhagen School interpreted the wave function as "probabilistic" nature [23]. Bohr rejected Einstein's insistence on causality by replacing the term "complementarity" with his own invention, "complementarity". Bohr rejects Einstein's insistence on causality, replacing it with his own invention of "complementarity" [5].

*Schrödinger ascribed a reality of the same kind that light waves possessed to the waves that he regards as the carriers of atomic processes by using the de Broglie procedure; he attempts "to construct wave packets (wave parcels) that have relatively small dimensions in all directions," and which can obviously represent the moving " and which can obviously represent the moving corpuscle directly*[4][6].

*Born and Heisenberg believe that an exact representation of processes in space and time is quite impossible and that one must then content oneself with presenting the relations between the observed quantities, which can only be interpreted as properties of the motions in the limiting classical cases [6]. Heisenberg, in contrast to Bohr, believed that the wave equation gave a causal, albeit probabilistic description of the free electron in configuration space*[1].

The wave function itself is a function of time and space, and if the "wave-function collapse" at the time of measurement is probabilistic evolution, with instantaneous nature, [3], neither time (Δt=0) nor spatial transition is required. then it is in conflict not only with the Special Relativity, but also with the Uncertainty Principle. Because the wave function represents some definite energy and momentum, which appear to be infinite when required to follow the Uncertainty Principle [7], ΔE*Δt>h and ΔP*Δx>h.

V: We must also be mindful of the fact that the amount of information about a completely random event. From a quantum measurement point of view, it is infinite, since the true probability event of going from a completely unknown state A before the measurement to a completely determined state B after the measurement is completely without any information to base it on‡.

VI: The Uncertainty Principle originated in Heisenberg's analysis of x-ray microscopy [8] and its mathematical derivation comes from the Fourier Transform [8][10]. E and t, P and x, are two pairs of commuting quantities [11]. While the interpretation of the Uncertainty Principle has been long debated [7][9], "Either the color of the light is measured precisely or the time of arrival of the light is measured precisely." This choice also puzzled Einstein [12], but because of its great convenience as an explanatory "tool", physics has extended it to the "generalized uncertainty principle " [13].

Is this tool not misused? Take for example a time-domain pulsed signal of width τ, which has a Stretch (Scaling Theorem) property with the frequency-domain Fourier transform [14], and a bandwidth in the frequency domain B ≈ 1/τ. This is the equivalent of the uncertainty relation¶, where the width in the time domain is inversely proportional to the width in the frequency domain. However, this relation is fixed for a definite pulse object, i.e., both τ and B are constant, and there is no problem of inaccuracy.

In physics, the uncertainty principle is usually explained in terms of single-slit diffraction [15]. Assuming that the width of the single slit is d, the distribution width (range) of the interference fringes can be analyzed when d is different. Describing the relationship between P and d in this way is equivalent to analyzing the forced interaction that occurs between the incident particle and d. The analysis of such experimental results is consistent with the Fourier transform. But for a fixed d, the distribution does not have any uncertainty. This situation is confirmed experimentally, "We are not free to trade off accuracy in the one at the expense of the other."[16].

The usual doubt lies in the diffraction distribution that appears when a single photon or a single electron is diffracted. This does look like a probabilistic event. But the probabilistic interpretation actually negates the Fourier transform process. If we consider a single particle as a wave packet with a phase parameter, and the phase is statistical when it encounters a single slit, then we can explain the "randomness" of the position of a single photon or a single electron on the screen without violating the Fourier transform at any time. This interpretation is similar to de Broglie's interpretation [17], which is in fact equivalent to Bohr's interpretation [18][19]. Considering the causal conflict of the probabilistic interpretation, the phase interpretation is more rational.

VII. The uncertainty principle is a "passive" principle, not an "active" principle. As long as the object is certain, it has a determinate expression. Everything is where it is expected to be, not this time in this place, but next time in another place.

**Our problems are**:

1) At observable level, energy-momentum conservation (that is, causality) is never broken. So, is it an active norm, or just a phenomenon?

2) Why is there a "probability" in the measurement process (wave packet collapse) [3]?

3) Does the probabilistic interpretation of the wave function conflict with the uncertainty principle? How can this be resolved?

4) Is the Uncertainty Principle indeed uncertain?

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**Notes**:

* Determinism here is a narrow sense of determinism, only for localized events. My personal attitude towards determinism in the broad sense (without distinguishing predictability, Fatalism, see [20] for a specialized analysis) is negative. Because, 1) we must note that complete prediction of all states is dependent on complete boundary conditions and initial conditions. Since all things are correlated, as soon as any kind of infinity exists, such as the spacetime scale of the universe, then the possibility of obtaining all boundary conditions is completely lost. 2) The physical equations of the upper levels can collapse by entering a singularity (undergoing a phase transition), which can lead to unpredictability results.

** Personal, non-professional opinion.

*** Energy conservation of independent wave functions is unquestionable, and it is debatable whether the interactions at the time of measurement obey local energy conservation [21].

† This is precisely the meaning of the Planck Constant h, the smallest unit of action. h itself is a constant of magnitude Js. For the photon, when h is coupled to time (frequency) and space (wavelength), there is energy E = hν,momentum P = h/λ.

‡ Thus, if a theory is to be based on "information", then it must completely reject the probabilistic interpretation of the wave function.

¶ In the field of signal analysis, this is also referred to by some as "The Uncertainty Principle", ΔxΔk=4π [22].

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**References**：

[1] Faye, J. (2019). "Copenhagen Interpretation of Quantum Mechanics." The Stanford Encyclopedia of Philosophy from <https://plato.stanford.edu/archives/win2019/entries/qm-copenhagen/>.

[2] Weinberg, S. (2020). Dreams of a Final Theory, Hunan Science and Technology Press.

[3] Bassi, A., K. Lochan, S. Satin, T. P. Singh and H. Ulbricht (2013). "Models of wave-function collapse, underlying theories, and experimental tests." Reviews of Modern Physics 85(2): 471.

[4] Schrödinger, E. (1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules." Physical Review 28(6): 1049-1070.

[5] Bohr, N. (1937). "Causality and complementarity." Philosophy of Science 4(3): 289-298.

[6] Born, M. (1926). "Quantum mechanics of collision processes." Uspekhi Fizich.

[7] Busch, P., T. Heinonen and P. Lahti (2007). "Heisenberg's uncertainty principle." Physics Reports 452(6): 155-176.

[8] Heisenberg, W. (1927). "Principle of indeterminacy." Z. Physik 43: 172-198. “不确定性原理”源论文。

[9] https://plato.stanford.edu/archives/sum2023/entries/qt-uncertainty/; 对不确定性原理更详细的历史介绍，其中包括了各种代表性的观点。

[10] Brown, L. M., A. Pais and B. Poppard (1995). Twentieth Centure Physics（I）, Science Press.

[11] Dirac, P. A. M. (2017). The Principles of Quantum Mechanics, China Machine Press.

[12] Pais, A. (1982). The Science and Life of Albert Einstein I

[13] Tawfik, A. N. and A. M. Diab (2015). "A review of the generalized uncertainty principle." Reports on Progress in Physics 78(12): 126001.

[15] 曾谨言 (2013). 量子力学（QM）, Science Press.

[16] Williams, B. G. (1984). "Compton scattering and Heisenberg's microscope revisited." American Journal of Physics 52(5): 425-430.

Hofer, W. A. (2012). "Heisenberg, uncertainty, and the scanning tunneling microscope." Frontiers of Physics 7(2): 218-222.

Prasad, N. and C. Roychoudhuri (2011). "Microscope and spectroscope results are not limited by Heisenberg's Uncertainty Principle!" Proceedings of SPIE-The International Society for Optical Engineering 8121.

[17] De Broglie, L. and J. A. E. Silva (1968). "Interpretation of a Recent Experiment on Interference of Photon Beams." Physical Review 172(5): 1284-1285.

[18] Cushing, J. T. (1994). Quantum mechanics: historical contingency and the Copenhagen hegemony, University of Chicago Press.

[19] Saunders, S. (2005). "Complementarity and scientific rationality." Foundations of Physics 35: 417-447.

[21] Carroll, S. M. and J. Lodman (2021). "Energy non-conservation in quantum mechanics." Foundations of Physics 51(4): 83.

[23] Born, M. (1955). "Statistical Interpretation of Quantum Mechanics." Science 122(3172): 675-679.

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In quantum mechanics, the Schrödinger equation calculated wavefunctions with a wave structure over space and changing over time. The Copenhagen interpretation, namely Born‘s interpretation states that the square modulus of the wavefunction represents the probability density function of the particle over space and time. Thus, there will be a distribution of the particle over space because we know particles are moving in the system and may favor some locations.

This is a very confusing explanation that several founders of Quantum Mechanics including Schrödinger himself, Einstein, and de Broglie have formally expressed disagreement.

I have been teaching undergraduate quantum chemistry for several years and also felt difficult to explain the probability density function why there are nodes in the solution where particles will never show up with no particular reason to avoid those places. I have been trying to come up with a different explanation of the wavefunctions with a preprint firstly posted on ChemRxiv in 4/2021. Since then I have been thinking on it and working on revisions while teaching quantum again in the past few years.

DOI: 10.26434/chemrxiv-2022-xn4t8-v17

It reaches a very surprising conclusion that the wavefunction has nothing to do with statistics as Schrödinger himself has argued many times including the famous Schrödinger’s cat thought experiment.

I recently posted the preprint in RG. Please take a read and comments are welcome. I will be teaching quantum again next semester now I have even more difficulties since I have lost beliefs on the classical interpretation.

Suppose A is a set measurable in the Caratheodory sense such for n in the integers, A is a subset of R

^{n}, and function f:A->RAfter reading the preliminary definitions in section 1.2 of the attachment where, e.g.,

*a**pre-structure*is a sequence of sets whose union equals A and each term of the sequence has a positive uniform probability measure; how do we answer the following question in section 2?Does there exist a unique extension (or method constructively defining a unique extension) of the expected value of f when the value’s finite, using the uniform probability measure on sets measurable in the Caratheodory sense, such we replace f with infinite or undefined expected values with f defined on a chosen

*pre-structure*depending on A where:- The expected value of f on each term of the pre-structure is finite
- The
*pre-structure**converges uniformly*to A - The
*pre-structure**converges uniformly*to A at a*linear*or*superlinear*rate to that of other non-equivalent*pre-structures*of A which satisfies 1. and 2. - The
*generalized expected value of f*on the*pre-structure*(an extension of def. 3 to answer the full question) satisfies 1., 2., and 3. and is unique & finite. - A choice function is defined that chooses a
*pre-structure*from A that satisfies 1., 2., 3., and 4. for the largest*possible*subset of R^{A}. - If there is more than one choice function that satisfies 1., 2., 3., 4. and 5., we choose the choice function with the "simplest form", meaning for a general
*pre-structure*of A (see def. 2), when each choice function is fully expanded, we take the choice function with the fewest variables/numbers (excluding those with quantifiers).

How do we answer this question?

(See sections 3.1 & 3.3 in the attachment for an idea of what an answer would look like)

**Edit:**Made changes to section 3.5 (b) since it was nearly impossible to read. Hopefully, the new version is much easier to process.

Hello!

I have a few questions about a TDDFT calculation that I ran ( # td b3lyp/6-31g(d,p) scrf=(iefpcm,solvent=chloroform) guess=mix) and when I calculation the % probability of some of the excitation states I am getting >100%. What I remember from statistics is that we cannot actually have >100% probability so I am trying to figure out why I have that occurring in my data.

I calculated %probability by 2*(coefficient^2). I have included one of data's oscillator strength information below.

"Excitation energies and oscillator strengths:

Excited State 1:

**2.047-A**0.5492 eV 2257.37 nm f=0.1453 <S**2>=0.797 339B -> 341B 0.20758 (8.62%)

340B -> 341B 0.97366 (189.6%)

This state for optimization and/or second-order correction.

Total Energy, E(TD-HF/TD-DFT) = -6185.76906590

Copying the excited state density for this state as the 1-particle RhoCI density.

Excited State 2:

**2.048-A**0.6312 eV 1964.25 nm f=0.0730 <S**2>=0.798 339B -> 341B 0.97645 (190.69%)

340B -> 341B -0.20706 (8.57%)

Excited State 3:

**2.037-A**0.7499 eV 1653.42 nm f=0.0000 <S**2>=0.787 331B -> 341B 0.98349 (193.45%)

SavETr: write IOETrn= 770 NScale= 10 NData= 16 NLR=1 NState= 3 LETran= 64.

Hyperfine terms turned off by default for NAtoms > 100."

The other three questions I have are:

- what the ####-A means (in bold above) as I have some calculations with various numbers-A and others that have singlet-A. (ZnbisPEB file)
- I obtained a negative wavelength what should I do? I have already read a question on here about something similar, but the only suggestion was to remove the +, which I do not have in my initial gjf file. Should I solve for more states or completely eliminate (d,p)? (Trimer file)
- In another calculation I obtained a negative oscillator strength which I know from some web searches is not theoretically possible and indicates that there is a lower energy state (is that correct?) - how would I fix that? I have included it below, the same basis set as above is used. (1ZnTrimer file)

"Excitation energies and oscillator strengths:

Excited State 1: 4.010-A -0.2239 eV -5538.35 nm f=-0.0004 <S**2>=3.771"

Any clarification would be super helpful. I have also included the out files for the three compounds I am talking about.

Thank you so much!

Could any expert try to examine our novel approach for multi-objective optimization?

The brand new approch was entitled "Probability - based multi - objective optimization for material selection", and published by Springer available at https://link.springer.com/book/9789811933509,

DOI: 10.1007/978-981-19-3351-6.

P(y)=Integral( P(x|y)*P(y)*dx)

function is above if I didnt write wrongly. P(x|y) is conditional probability and it is known but P(y) is not known, thanks.

there may be an iterative solution but is there any analytical solution.

My lab recently got donated a 5500xl Genetic Analyzer from Applied Biosystems. However they are discontinuing the official reagents for this system come Dec 31, 2017; which is probably why it was given for free.

So I am wondering if anyone can offer help to get this machine running on generic reagents, or any tips/hints/advice; or even if it's worth the effort.

Basically it would be nice to get it sequencing, but if that can't be done are there any salvageable parts (for instance I know it has a high def microscope and a precise positioning system) ?

Here is a link to the machine we have:

We have all the accessory machines that go with it.

Thanks.

Suppose that we have a two-component series system, what is the probability of failure of two components at the same time?

*** both components' failure times are continuous random variables,

*** Is it important that they follow the same distribution or the different ones or the same distribution with different parameters?

In more detail, which conditions should be held that if X is a continuous random, then f(X) is also a continuous RV?

More specially, if X and Y are two continuous RVs, is X-Y a continuous RV?

Bests

If for example the position of an electron in a one-dimensional box is measured at A (give and take the uncertainty), then the probability of detecting the particle at any position B at a classical distance from A becomes zero instantaneously.

In other words, the "probability information" appears to be communicated from A to B faster than light.

The underlying argument would be virtually the same as in EPR. The question might be generalized as follows: as the probability of detecting a particle within an arbitrarily small interval is not arbitrarily small, this means that quantum mechanics must be incomplete.

Yet another formulation: are the collapse of the wave function and quantum entanglement two manifestations of the same principle?

It should be relatively easy to devise a Bell-like theorem and experiment to verify "spooky action" in the collapse of the wave function across a given classical interval.

I want to draw a graph between predicted probabilities vs observed probabilities. For predicted probabilities I use this “R” code (see below). Is this code ok or not ?.

Could any tell me, how can I get the observed probabilities and draw a graph between predicted and observed probability.

analysis10<-glm(Response~ Strain + Temp + Time + Conc.Log10

+ Strain:Conc.Log1+ Temp:Time

,family=binomial(link=logit),data=df)

predicted_probs = data.frame(probs = predict(analysis10, type="response"))

I have attached that data file

I want to see the distribution of an endogenous protein(hereafter as A), and I followed the protocol from Axis-Shield (http://www.axis-shield-density-gradient-media.com/S21.pdf). In order to gain stronger signals, I tried small ultracentrifuge tube (SW60, 4ml).

In this protocol, Golgi is enrich in #1~3, ER is #9~12. But in my experiments,the enrichment of ER (Marker using Calnexin) is usually failed (#3~12), while Golgi (marker using GM130) is good (#1~3).

Here are some questions :

1. the amount of protein loaded on gradient: Should it be considered? I mean, is the capacity of the gradient need to be think out? does it effect the fraction efficiency?

2. Is it necessary to use large tube (12ml)? Previously, to gain stronger signals of A, I switched to smaller tube (4ml). I have searched many papers, and some of them use small tube for fractionation. (Probably they use less protein for loading)?

Thanks a lot for answering

Why the Chi-square cannot be less than or equal to 1 ?

I have camera traps data which was deployed at several sites (islands). The data consist of only N (abundance; independent image) of each species at their respected islands. No occupancy modelling could be run as I do not have the habitat data. Is it possible to calculate the occupancy and probability without the temporal data/repeated sampling (the week the species was detected)? Or would calculating the Naive Occupancy would do? Furthermore, does Occupancy and Probability has a range of high and low?

probability and fuzzy numbers are ranged between 0 to 1. both are explaned in similar manner. what is the crisp difference between these terms?

Hi everyone,

In engineering design, there are usually only a few data points or low order moments, so it is meaningful to fit a relatively accurate probability density function to guide engineering design. What are the methods of fitting probability density functions through small amounts of data or low order statistical moments?

Best regards

Tao Wang

Imagine there is a surface, with points randomly spread all over it. We know the surface area S, and the number of points N, therefore we also know the point density "p".

If I blindly draw a square/rectangle (area A) over such surface, what is the probability it'll encompass at least one of those points?

P.s.: I need to solve this "puzzle" as part of a random-walk problem, where a "searcher" looks for targets in a 2D space. I'll use it to calculate the probability the searcher has of finding a target at each one of his steps.

Thank you!

Dear colleagues.

In the following question i try to extend the concept of characters in group theory to a wilder class of functions. A character on a group G is a group homomorphism $\phi:G \to S^1$.

For every character $\phi=X+iY$ on a group $G$, we have $Cov(X.Y)=0$.

This is a motivation to consider all $\phi=X+iY: G\to S^1$ with $Cov(X,Y)=0$.

Please see this post:

So this question can be a starting point to translate some concepts in geometric group theory or theory of bamenability of groups in terms of notations and terminologies in statistics and probability theory.

Do you have any ideas, suggestions or comments?

Thank you

For example, dirchlet and multinomial distribution were conjugated. We want to know the probability of variable A occurs with B. Therefore, we train two probability model with enough samples and computational power. First model based on dirchlet distribution. Second model based on multinomial distributions. When we infer the parameter of dirchlet-multinomial and multinomial distributions, will the accuracy of models be different?

I need help with Queuing theory, easy explanation to M/M/C?

What are the parameters of the M/M/C queuing model?

Hi, everyone

In relation with the statistical power analysis, the relationship between effect size and sample size has crucial aspects, which bring me to a point that, I think, most of the time, this sample size decision makes me feel confusing. Let me ask something about it! I've been working on rodents, and as far as I know, a prior power analysis based on an effect size estimate is very useful in deciding of sample size. When it comes to experimental animal studies, providing the animal refinement is a must for researchers, therefore it would be highly anticipated for those researchers to reduce the number of animals for each group, just to a level which can give adequate precision for refraining from type-2 error. If effect size obtained from previous studies prior to your study, then it's much easier to estimate. However, most of the papers don't provide any useful information neither on means and standard deviations nor on effect sizes. Thus it makes it harder to make an estimate without a plot study.
So, in my case, when taken into account the effect size which I've calculated using previous similar studies, sample size per group (4 groups, total sample size = 40 ) should be around 10 for statistical power (0.80). In this case, what do you suggest about the robustness of checking residuals or visual assessments using Q-Q plots or other approaches when the sample size is small (<10) ?

Kind regards,

I suspect this is a well-worn topic in science education and psychology, but these are fields I don't know well. I'd like a source or two to support my sense that probability/statistics are hard for people to understand and correctly interpret because they defy "the way our minds work" (to put it crudely). Any suggestions?

I have a large set of sampled data. How can I plot a normal pdf from the available data set?

In R-studio, there are many commands of Gumbel package. Arguments are also different.

I`m asking about the alpha parameter of the Copula which must be greater than 1. If this is the one used to plot the probability paper, how can I choose the value of alpha?

I am considering to distribute N-kinds of different parts among M-different countries and I wan to know the "most probable" pattern of distribution. My question is in fact ambiguous, because I am not very sure how I can distinguish types or patterns.

Let me give an example. If I were to distribute 3 kinds of parts to 3 countries, the set of all distribution is given by a set

{aaa, aab, aac, aba, abb, abc aca, acb, acc, baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc, caa, cab, cac, cba, cbb, cbc, cca, ccb, ccc}.

The number of elements is of course 33 = 27. I may distinguish three types of patterns:

(1) One country receives all parts:

aaa, bbb, ccc 3 cases

(2) One country receives 2 parts and another country receives 1 part:

aab, aac, aba, abb, aca, acc, baa, bab, bba, bbc, bcb, caa, cac, cbb, cbc, cca, ccb 17 cases

(3) Each county rceives one part respectively:

abc, acb, bac, bca, cab, cba 6 cases

These types may correspond to a partition of integer 3 with the condition that (a) number of summands must not exceed 3 (in general M). In fact, 3 have three partitions:

3, 2+1, 1+1+1

In the above case of 3×3, the number of types was the number of partitions of 3 (which is often noted p(n)). But I have to consider the case when M is smaller than N.

If I am right, the number of "different types" of distributions is the number of partitions of N with the number of summands less than M+1. Let us denote it as

p*(N, M) = p( N | the number of summands must not exceed M. )

N.B. * is added in order to avoid confusion with p(N, M), wwhich is the number of partitions with summands smaller than M+1.

Now,

**my question is the following**:*Which type (a partition among p*(N, M)) has the greatest number of distributions?*

Are there any results already known? If so, would you kindly teach me a paper or a book that explains the results and how to approach to the question?

A typical case that I want to know is N = 100, M = 10. In this simple case, is it most probable that each country receives 10 parts? But, I am also interested to cases when M and N are small, for example when M and N is less than 10.

Do we have a mathematical formula to compute the p-value of an observation from the Dirichlet distribution in exact sense at https://en.wikipedia.org/wiki/Exact_test?

I have a data which consists of an excess of zero counts. The independent variables are number of tree, diameter at breast height and basal area, and the dependent variable (predictors) is number of recruits (with many zero counts).

So, I want to use Zero-inflated negative binomial model and Hurdle negative binomial model to analyze. My problem is I do not know the code of these models in R package.

How can I calculate and report degrees of freedom for repeated mesure ANOVA?

I have 48 observations N=48 and 2 factors of 3(P) and 8(LA) levels.

I calculate degrees of freedom as follows:

dF P = a-1= 2

df LA = b-1= 7

df LA*P =(a-1)(b-1)= 14

Error dF P = (a-1) (N-1) = 94

Error dF LA = (b-1) (N-1) = 329

Error dF P*LA = (a-1)(b-1)(N-1) = 658

My JASP analysis gave me these results:

Within Subjects Effects

Cases Sum of Squares df Mean Square F p η²

P 1.927 2 0.964 33.9 < .001 0.120

P*LA 8.450 14 0.604 21.2 < .001 0.528

Residuals 0.454 16 0.028

Can I write P : F(2,14)= 33.9

and P*LA: F(14, 658) =21.2 ???

Or is it P: F(2, 16)=33.9

P*LA: F(14, 16) =21.2 ???

Thanks to anyone who would like to answer

Please consider a set of pairs of probability measures (P, Q) with given means (m_P, m_Q) and variances (v_P, v_Q).

For the relative entropy (KL-divergence) and the chi-square divergence, a pair of probability measures defined on the common two-element set (u_1, u_2) attains the lower bound.

Regarding general f-divergence, what is the condition of f such that a pair of probability measures defined on the common two-element set attains the lower bound ?

Intuitively, I think that the divergence between localized probability measures seems to be smaller.

Thank you for taking your time.

How to prove or where to find the integral inequality (3.3) involving the Laplace transforms, as showed in the pictures here? Is the integral inequality (3.3) valid for some general function $h(t)$ which is increasing and non-negative on the right semi-axis? Is the integral inequality (3.3) a special case of some general inequality? I have proved that the special function $h(t)$ has some properties in Lemma 2.2, but I haven't prove the integral inequality (3.3) yet. Wish you help me prove (3.3) for the special function $h(t)$ in Lemma 2.2 in the pictures.

**Hello dear,**

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In Garman's inventory model, buying order and selling order are poisson process with order size = 1. Buying price and selling price are denoted by pb and ps, that is, the market maker gets pb when she sells a stock to the others, and spends ps to buy a stock from the others.

Garman than calculates the probability of the inventory of the market maker, says Q(k, t+dt) = probability to get 1 dollar x Q(k-1, t) + probability to lose 1 dollar x Q(k+1, t) + no buying or selling order x Q(k, t), where Q(k, t+dt) = probability to have k money at time t+dt.

In the above equation, I think Garman had split the money received and loss by buying or selling a shock in many sub-poisson process, otherwise, getting 1 dollar or losing 1 dollar are impossible, as market maker receive pb dollar and loses ps dollar in each order, but not 1 dollar. Do my statement correct? Thank you very much.

I have a list of chromosomes, say A, B, C, and D. The respective fitness values are 1, 2, 3, and 4. The chromosomes with higher fitness values (C and D) are more likely to be selected for the parent in the next generation. Therefore, how to assign probability in MATLAB such that C and D get a higher probability for parent selection?

Suppose we have statistics N(m1, m2), where m1 is the value of the first factor, m2 is the value of the second factor, N(m1, m2) is the number of observations corresponding to the values of factors m1 and m2. In this case, the probability P(m1, m2) = N(m1, m2) /K, where K is the total number of observations. In real situations, detailed statistics N(m1, m2) is often unavailable, and only the normalized marginal values S1(m1) and S2(m2) are known, where S1(m1) is the normalized total number of observations corresponding to the value m1 of the first factor and S2(m2) is the normalized total number of observations corresponding to the value m2 of the second factor. In this case P1(m1) = S1(m1)/K and P2(m2) = S2(m2)/K. It is clear that based on P1(m1) and P2(m2) it is impossible to calculate the exact value of P(m1, m2). But how to do this approximately with the best confidence? Thanks in advance for any advice.

Dear all. The

**Normal distribution**(or Gaussian) is mostly used in statistics, natural science and engineering. Their importance is linked with the**Central Limit Theorem**. Is there any ideas how to predict the numbers and parameters of thos Gaussians ? Or any efficient deterministic tool to decompose Gaussian to a finite sum of**Gaussian basic functions**with parameter estimations ? Thank you in advance.Dear all. The Gaussian function is mostly used in statistics, physics and engineering. Some examples include:

1. Gaussian functions are the

**Green's function for**the homogeneous and isotropic2. The convolution of a function with a Gaussian is also known as a

**Weierstrass transform**3. A Gaussian function is the wave function of the ground state of the

**quantum harmonic oscillator**3. The

**atomic**and**molecular orbitals**used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals4. Gaussian functions are also associated with the

**vacuum state**in quantum field theory5. Gaussian beams are used in optical systems,

**microwave systems and lasers**6. Gaussian functions are used to define some types activation function of artificial

**neural networks**7.

**Simple cell response i**n primary visual cortex has a Gaussian function modeled by a sine wave8. Fourier transforms of Gaussian is a Gaussian

10. Easy and efficient approximation for signals analysis and fitting (Gaussian process, gaussian mixture model, kalman estimator , ...)

11. Discribe the Shape of the

**UV−Visible Absorption**12. Used in Time-frequency analysis (

**Gabor Transform**)13.

**Central Limit Theorem**(**CLT**) : Sum of independent random variables tends toward a normal distribution14. The

**Gaussian function****serves well in****molecular physics**, where the number of particles is closed to the**Avogadro number NA = 6.02214076×1023 mol−1**( N**A**is defined as the number of particles that are contained in one mole)15. ...

Why Gaussian is everywhere ?

I was working on the C++ implementation of Dirichlet Distribution for the past few days. Everything is smooth, but I am not able to deduce the CDF (Cumulative Distribution Function) of Dirichlet distribution. Unfortunately, neither Wikipedia nor WolframAlpha shows the CDF formula for Dirichlet Distribution.

Forgetting the context for a second, the overall question is how to compare data that is expressed in a probability.

Scenario 1: Let us say there are two events A and B. The rules are:

- A (union) B = 1

- A (intersection) B = 0

- Probability of A or B is dependent on a dollar amount. Depending on the amount, the probability either A or B happens changes. For e.g. @20,000 chance of A is 80%, then B is 20%.

Scenario 2: we have A, B, and C.

- A (union) B (union) C = 1

- A (intersection) B or C = 0

- Probability dependent on dollar amount. Same as above.

- A and B in scenarios 1 and 2 are same but their probabilities of happening are changed due to the introduction of C.

QUESTION: How can I compare the probability of the events in these two scenarios?

Possible solutions I was thinking of:

1) A is X times as likely to happen as B, then I could plot all events as a factor of B on the same graph to get a sense of how likely all events are compared to a common denominator (event B)

2) Could also get a "cumulative" probability of each event as area under the curve and express as a % or ratio. So if A occupies 80% of the area under the curve, then B should be 20%, so overall A is four times as likely, and similarly in scenario 2.

3) Maybe the way to compare is to take the complement of each event separately, and express as a percentage at each point and graph them.

Any help is greatly appreciated. Please refer to attached pic for some visual understanding of the question as well. I am making a lot of assumptions, which are not true (as concerned with the graphs etc), but theoretically, I am interested in knowing. Thank you!

How to find the distance distribution of a random point in a cluster from the origin? I have uniformly distributed cluster heads following the Poisson point process and users are deployed around the cluster head, uniformly, following the Poisson process. I want to compute the distance distribution between a random point in the cluster and the origin. I have attached the image as well, where 'd1', 'd2', and 'theta' are Random Variables. I want to find the distribution of 'r'.

hello dears, I have just interested in IFRS9 ECL models. I have three question and I appreciate all answers.

1) which models are best for pd, LGD & EAD calculation when I have scarce (about 5-7 years quarterly data) data?

2) can I calculate lifetime PD without macroeconomic variables and then add macro effects?

3) when I use transition matrix approach how have to estimate "stage 2" for earlier period, when IFRS9 was not valid and there was not any classification by stages.