CN101630459B - Coupled normal distribution probability demonstration instrument - Google Patents

Abstract

The invention discloses a coupled normal distribution probability demonstration instrument, improves the normal distribution demonstration instrument Galton pinboard, and is used for visually demonstrating normal distribution coupling relation. The construction scheme is as follows: regular hexagonal prisms form nails which are vertically adhered on front and back substrates, the adjacent surfaceof two adjacent regular hexagonal prisms and front and back substrates form clearance which constitutes a globule running channel, thus ensuring that the probability of globules falling into a left pipeline or a right pipeline after touching a nail is equal and the globules fall off along the channel and vertically touch the nail edge of the next layer. The substrate is provided with a globule feeding device, a regular hexagonal prism honeycomb array, globule catching tubes and a globule storage box; the globule feeding device is constructed into different globule feeding proportions of two paths of the globules according to binomial distribution principle, the regular hexagonal prism honeycomb array is formed by coupling two independent normal distribution models, and channel of the lastlayer are the globule catching tubes, all globule catching tubes form an array face which visually demonstrate coupled normal distribution probability graphics of the globules entering the regular hexagonal prism honeycomb array. The invention materializes abstracts concept to material objects, which plays an active role for the instruction of probability theory.

Description

Coupled normal distribution probability demonstration instrument

Technical field

The invention belongs to technical field of teaching instruments, particularly a kind of probability demonstrator can be used in the teaching of theory of probability, is used for demonstrating the coupling situation between the normal distribution.

Background technology

People usually can run into some variablees in real work and life, their distribution is similar to normal distribution.Such chance phenomenon has been described in normal distribution: if the value of a stochastic variable only is subjected to numerous, small stochastic factor, and the independent role of each factor all can be ignored, and this stochastic variable is with regard to Normal Distribution so.For example: the bullet drop point is to the lateral deviation and the along track bias of central point during shooting, be subjected to influence of various factors such as at that time weather, wind direction, ejaculator's personality and psychology, these are all very small, so that can ignore, but, the summation of these influences but shows significantly by firing error, the firing error Normal Distribution.

Normal distribution plays an important role in theory of probability." two broad in the middle is little " said in the proverb is exactly a character of normal distribution.In practice, we usually will consider total influence that many enchancement factors produce.Generally, can regard many small as, independently the variable of total consequence of enchancement factor can be thought Normal Distribution.In history, normal distribution is then found when studying theory of errors by Gauss (Gauss) at first.

Present existing normal distribution demonstrator mainly is a Galton nail-plate model, and its shape as shown in Figure 8.Galton nail-plate structure of models is: nail equates that with distance between the nail promptly each hole is big or small identical, and than the slightly larger in diameter of bead some; From horizontal direction, the nail on the edge, each nail all is in the centre position in last layer or the hole that one deck is relative with it down; After bead was met certain nail, it only can be rolled in that hole on this nail left side or the right, and the probability in two holes equates about entering, and all is 1/2.

According to the central limit theorem in the theory of probability: establish independent random variable row { ξ _kVariance D ξ _kLimited, 0＜D ξ _k＜∞, and make a _k=E ξ _k, ${\mathrm{<figure-callout\; id="2"\; label="b\; k"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">b</figure-callout>}}_k^{}={\mathrm{D\&xi;}}_k,$ ${\mathrm{<figure-callout\; id="2"\; label="b\; n"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">b</figure-callout>}}_n^{}=\underset{k=1}{\overset{\mathrm{<figure-callout\; id="2"\; label="n\; b\; k"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">n</figure-callout>}}{\mathrm{\&Sigma;}}}{b}_k^{}{}_2^{},$ E ξ _kBe ξ _k(k=1,2 ...) expectation, then have:

$\underset{n\&RightArrow;\&infin;}{\lim }P(\frac{1}{B_n}\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}({\mathrm{\&xi;}}_k-a_k)\&le;x)=\frac{1}{\sqrt{2\mathrm{\&pi;}}}{\&Integral;}_{-\&infin;}^x{e}^{-\frac{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">z</figure-callout>}}^2}{2}}\mathrm{dz}$

About x ∈ R ₁All set up.

If independent random variable row { ξ _kHave identical distribution, then have:

$\underset{n\&RightArrow;\&infin;}{\lim }P(\frac{1}{b_1\sqrt{n}}\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}({\mathrm{\&xi;}}_k-a_1)\&le;x)=\frac{1}{\sqrt{2\mathrm{\&pi;}}}{\&Integral;}_{-\&infin;}^x{e}^{-\frac{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">z</figure-callout>}}^2}{2}}\mathrm{dz}$

Set up, wherein a _k=a ₁, $b_{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">k</figure-callout>}}^2=b_1^2.$

It is the fundamental function of normal distribution N (0,1).Following formula explanation: for each x ∈ (∞, ∞), random variable

Distribution function be tending towards the distribution function of normal distribution N (0,1) equably.For Galton nail-plate model, we define with the distribution random variable and are listed as:

So ξ _kDensity matrix be $\left(\begin{array}{cc}-1& 1\\ \frac{1}{2}& \frac{1}{2}\end{array}\right),$ And ${\mathrm{\&eta;}}_n=\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_k$ Represent to bump the groove numbering that nail back ball is fallen into for the n time, its expectation and variance are: a _k=E ξ _k=0, b _k=D ξ _k=1, then have:

$\underset{n\&RightArrow;\&infin;}{\mathrm{lin}}P(\frac{{\mathrm{\&eta;}}_n}{\sqrt{n}}\&le;x)=\frac{1}{\sqrt{2\mathrm{\&pi;}}}{\&Integral;}_{-\&infin;}^x{e}^{-\frac{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">z</figure-callout>}}^2}{2}}\mathrm{dz}$

This expression is when n is fully big

Distribution be similar to N (0,1), that is η _nDistribution be similar to

Get by following formula:

$P(-l{\&le;\mathrm{\&eta;}}_n\&le;l)=P(\frac{-l}{n}\&le;\frac{{\mathrm{\&eta;}}_n}{n}\&le;\frac{l}{n})\&ap;\frac{1}{\sqrt{2\mathrm{\&pi;}}}{\&Integral;}_{\frac{-l}{\sqrt{n}}}^{\frac{l}{\sqrt{n}}}{e}^{-\frac{{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">z</figure-callout>}}^2}{2}}\mathrm{dz}$

As axis, the groove of corresponding model below is numbered 0 with first a unique nail of arranging, and it is-2 that its first left groove is numbered-1, the second, by that analogy; It is 2 that first groove of the right is numbered 1, the second, by that analogy.Bead tumbles l groove explanation: in the process of bumping n nail, the number of times that rolls to the right manys l time than rolling to the left side; In like manner bead tumbles the-l ball receiving groove explanation: in the process of bumping n nail, the number of times that rolls to the left side manys l time than rolling to the right.In view of the above, can utilize gaussian distribution table to predict bead number in each groove.

When n is odd number, η _nValue only may be odd number, the numbering of groove must change into :-n ...-5 ,-3 ,-1,1,3,5 ... n; And when n is even number, η _nValue only may be even number, the numbering of groove must change into :-n ...-4 ,-2,0,2,4 ... n.

Adopt the binomial distribution principle to analyze.When n → ∞, binomial distribution is exactly normal distribution.Binomial distribution derives from shellfish and makes great efforts (Bernoulli) test, is a kind of discrete type probability distribution.Its content is: establish test E and have only two possible outcomes: incident A takes place and incident A, i.e. A do not take place.Probability P (the A)=p that takes place, the probability P that does not take place (A)=q=1-p, (0＜p＜1).To test E and carry out n time independently, the independent experiment that then claims this a string repetition is that the heavy shellfish of n is made great efforts test.Representing the number of times that incident A takes place in the heavy shellfish effort of the n test with X, is a stochastic variable, and its all possible value is: 1,2,3 ... n.Each time test is separate, thus incident A the k of appointment (0≤k≤n) take place in the inferior test, the probability that other n-k time test do not take place is:

$P\{X=k\}=C_n^k{p}^k{q}^{n-k},k=\mathrm{0,1,2},...n$

In the Galton model, establish bead and bump a nail and roll to the right to incident A takes place one time, to rolling left, be incident A to not taking place, the probability that takes place and do not take place is 1/2, and the nail of establishing in the model has the n layer altogether, is equivalent to do altogether n test.According to symmetry, we only study the bead number of an axis side to the right, left side and its symmetry.Still in top mode groove is numbered: if n is odd number, the nail that first row is unique is as axis, under do not have groove (this is by due to the symmetry of structure), its right groove label respectively is: 1,3,5 ... n; If n is even number, will first row unique nail is as axis, and its pairing groove is numbered 0, and the right groove label respectively is: 2,4,6 ... n.The probability that bead falls into l groove is: $P\{X=\frac{n+l}{2}\}=C_n^{\frac{n+l}{2}}{\left(\frac{1}{2}\right)}^{\frac{n+l}{2}}{\left(\frac{1}{2}\right)}^{\frac{n-\mathrm{<figure-callout\; id="2"\; label="l"\; filenames="H2009100236386E00011.png,H2009100236386E00052.png"\; state="\{\{state\}\}">l</figure-callout>}}{2}}=\frac{n!}{2^n\left(\frac{n+l}{2}\right)!\left(\frac{n-l}{2}\right)!}$

Bead number in the Galton nail-plate test groove distributes can correctly demonstrate normal distribution, the teaching visual pattern more that makes normal distribution.And now, Galton nail-plate model has obtained widespread use as instruments used for education.

Usually some interesting phenomenon can occur in real life, for example in general the total marks of the examination of certain class are the situations that roughly meets a normal distribution.The classmate who promptly secures satisfactory grades and hang down branch is not a lot, and it is in the majority to obtain the classmate of medium achievement, that is to say that near the classmate average mark is maximum.Also can to occur the classmate that does well in and get poor results in the class in practice all more yet understand according to us, is that the medium classmate of achievement is on the low side on the contrary.As if being that horizontal ordinate is ordinate with the number with the achievement, the curve that obtains just approaches the distribution of two peak values like this, similarly is two results that the different source normal distributions interactions (coupling) of average produce.The distribution of these two parts is because the power (degree of tightness of coupling) of effect can a series of variations of generation each other.Loosened such as the classmate who succeeds in school that own achievement glides or the classmate who gets poor results rises by making great efforts achievement.So just be necessary also meaningful research some contents about the coupling normal distribution.

The popularity of using in real life along with two interactional normal distributions constantly increases, and is not enough with abstract theoretical cogency only, and is not easy to see intuitively its existence, thereby be unfavorable for being understood by the student and accepting in teaching.

Yet influence each other (coupling) between two normal distributions can not be clearly reacted in the nail-plate test of adopting the Galton model; Remove this, structurally also there is weak point in Galton model itself: the probability that (1) bead falls into two holes about nail equates to be guaranteed by the verticality between nail and the plate, nail was easy to after bead bumps nail, and along with the increase of test number (TN), test accuracy is more and more lower; (2) two holes were not vertically to remove to bump next nail about bead bumped and falls into after the nail, even this nail of one deck equates with the probability that the strict vertical also very difficult assurance of plate falls into two holes down.

Therefore need improve the Galton model, work out an instruments used for education device that can reflect coupling normal distribution form, in order to the authenticity of demonstrating and explanation coupling normal distribution exists.

Summary of the invention

The technical problem to be solved in the present invention is to overcome test accuracy difference that the structural defective of Galton demonstrating model caused and the deficiency that is difficult for the coupled relation between two normal distributions of clear reaction, design a kind of coupled normal distribution probability demonstration instrument, be used to realize ocular demonstration to the process that intercouples between the normal distribution, be convenient to the student and understand, improve the quality of teaching.

The scheme of the technical problem to be solved in the present invention is: improve the Galton model, change the nail on the Galton model nail-plate into positive six prisms; Utilize the binomial distribution principle make up to realize two beads sum ratios different go into ball device; With two independently normal distribution form coupling by modelling, design and produce reasonably and the probability demonstrator that is coupled between the normal distribution that can realize.Concrete constructing plan of the present invention: described nail is made of positive six prisms, positive six prisms vertically are bonded on two substrates in front and back, the space that the adjacent surface of adjacent two positive six prisms and front-back baseboard form constitutes bead operation " pipeline ", will with over against the seamed edge of positive six prisms of bead traffic direction as nail, following one deck positive six prismatic those ribs as " nail " are relative with the pipeline bottom center of last layer, guarantee bead bumped nail and fell into about the probability of two pipelines equate and fall along pipeline, vertically remove to bump down " nail " rib of positive six prisms of one deck; Described substrate is provided with into ball device, positive six prism honeycomb array, receive pipe and storage ball box, describedly go into ball device and be provided with positive six prisms, special-shaped block and removable positive six prisms by the binomial distribution principle, make up the difference of the goal ratio of two-way bead, as after different number colony between the basis that is coupled; Go into the left and right arms goal passage of ball device by symmetry and connect positive six prism honeycomb array, positive six prism honeycomb array by two independently normal distribution model be coupled to form, last one deck pipeline of positive six prismatic battle arrays is a long pipeline, constitute the pipe of receiving, all length of tube of receiving equate, arrange side by side, constitute a front, the directly perceived demonstration enters the bead coupled normal distribution probability figure of positive six prism honeycomb array.

According to coupled normal distribution probability demonstration instrument of the present invention, describedly go into to arrange four layers " nail " in the ball device: ground floor and the 3rd layer are respectively positive six prisms, the second layer left side still is positive six prisms, the right special-shaped block of arranging, the 4th layer is removable positive six prisms, probability is 1/2 to the left and right sides after bead bumps ground floor " nail ", begin the left and right sides from the second layer and form different ratios, through the 3rd layer of removable positive six prism that collides the 4th layer, directly cause the bead number of going into three kinds of different proportions of interruption-forming of two coupling bodies of the right and left to distribute.

According to coupled normal distribution probability demonstration instrument of the present invention, described removable positive six prisms are bonded on the horizontal bolt, correspondence position is opened a mouth on metacoxal plate, this opening size decides with the upper limit of positive six prismatic move left and right distances, horizontal bolt is fixed in the draw-in groove of opening both sides it can only laterally be moved, horizontal bolt moves and drives positive six prismatic moving simultaneously, horizontally two scales have been fixed, when these two scales overlap with edge of opening, represent that positive six prisms move to open centre, horizontal fastening can move left and right move on to edge of opening, and three kinds of states that control is scored, horizontal bolt move the back that finishes and fixed by the screw above the draw-in groove.

According to coupled normal distribution probability demonstration instrument of the present invention, two independently positive six prism honeycomb array of normal distribution begin to intersect at m layer pipeline, promptly the first coupling body rightmost pipeline and the second coupling body Far Left pipeline combine together, making two groups of beads enter by coupling layer in the pipeline of the other side institute " administration " is coupled, by the coupling number of plies be m as can be known, total (2m-1) individual pipe of receiving merges mutually, and the final pipe number of always receiving that forms is (2n-2m+1).

The present invention is according to binomial distribution, total derive the relation of the number of bead in the pipe of receiving after two coupling bodies interact with number of plies n and coupling cambium layer m:

Wherein, S is the bead sum, and the bead number that enters first coupling body is pS, (0＜p＜1), and the little nodule number that enters second coupling body then is (1-p) S, l ₁Be the symmetrical distribution numbering of first coupling body, span is | l ₁|≤[(n+1)/2]; l ₂Be the symmetrical distribution numbering of second coupling body, span is | l ₂|≤[is (n+1)/2], wherein [(n+1)/2] represents the smallest positive integral bigger than (n+1)/2, n is odd number l ₁And l ₂All get odd number, n is even number l ₁And l ₂All get even number; Satisfy condition | l ₁+ l ₂| the pipeline of≤2 (m-1) is the fusion pipeline after being coupled; Constitute coupling normal distribution figure by the bead post in each pipe of receiving.

The present invention has the following advantages compared with the prior art:

1) because the present invention designs and has produced coupling normal distribution demonstrator, filled up and lacked this blank of coupling normal distribution demonstrator in the probability teaching, make the teaching that is coupled between the normal distribution become more directly perceived and vivid, can intuitively observe the cognition stochastic process, and the meaning of statistics, abstract concept specific on the material object, is played a positive role to the teaching of theory of probability.

2) because the present invention has taken all factors into consideration the existing processes level in design process, therefore its manufacture difficulty is lower, and can obtain craft precision preferably, change nail into positive six prisms, overcome the deficiency that exists in the Galton model in the past, avoided some out of true factors, improved experiment effect, be easy in the probability teaching, carry out promotion and application.

3) because experimental result is represented in the present invention's distribution that what form by the bead number, and use long pipeline to store bead, therefore can be by observing model below pellet shapes becomes in the pipe distribution obtain the being coupled general picture of normal distribution of receiving, and it is formed impression the most intuitively.

4) because the present invention designs three kinds of different beads and counts ratio, can realize that a device presents three experimental results, make full use of resource, reach good effect.

Description of drawings

Fig. 1 is the main body view of coupling normal distribution demonstrator among the present invention

Fig. 2 is a coupling normal distribution demonstrator goal structure drawing of device among the present invention

Fig. 3 is coupling normal distribution demonstrator monomer structure figure among the present invention

Fig. 4 is the rear view of removable positive six prisms among the present invention

Fig. 5 is the structural drawing of storage ball box among the present invention

Fig. 6 is the unimodal synoptic diagram under the Galton nail-plate model synoptic diagram overcoupling situation

Fig. 7 (A) is a steel ball quantity Theoretical Calculation distribution plan in the long pipeline under 1: 1 ratio that is formed by opening

Fig. 7 (B) is a steel ball quantity Theoretical Calculation distribution plan in the long pipeline under 3: 5 ratios that formed by opening

Fig. 7 (C) is a steel ball quantity Theoretical Calculation distribution plan in the long pipeline under 7: 9 ratios that formed by opening

Fig. 8 is a Galton nail-plate model synoptic diagram

Fig. 9 is the left view of coupling normal distribution demonstrator among the present invention

Embodiment

The present invention is according to Galton Model Design and theoretical framework thereof, design the structure that is coupled between ball device and the normal distribution of going into that can form three kinds of different proportion goals, receive bead number in the pipe with the relational expression of total number of plies n and coupling cambium layer m according to each numbering, draw the notional result of the medium and small nodule number order distribution situation of pipe of receiving at last, design rational coupling normal distribution demonstrator.

The specific embodiment of the present invention is as follows:

As shown in Figure 1, demonstrator agent structure of the present invention on substrate 12, set into ball device 1, positive six prism honeycomb array 2, receive pipe 3 and storage ball box 4.In conjunction with Fig. 3, go into ball device 1 and be provided with positive six prisms 5, special-shaped block 1-1 and removable positive six prismatic 1-2 by the binomial distribution principle, be used to make up the difference of two-way bead goal ratio, as after the basis that is coupled between the different number colony; Go into the left and right arms goal passage of ball device 1 by symmetry and connect positive six prism honeycomb array 2, positive six prism honeycomb array by two independently positive six prisms of normal distribution " nail " model be coupled to form.Pipeline is made as long pipeline below last one deck " nail " of positive six prismatic gusts 2, manages 3 as receiving and stores the bead that receives, and the equal in length of the pipe 3 of respectively receiving, length reach and can number the length of managing interior bead by complete arrangement peak value; For the ease of forming visual impression, all pipes 3 of receiving constitute a front, and the different pellet shapes of quantity on the front in each pipe becomes goalpost, the bead coupled normal distribution probability figure that shows by positive six prism honeycomb array 2 directly perceived.A storage ball box 4 is set below each pipe 3 of receiving, is used for cooperating forming to receive and manages bead number in goalpost and each pipe of receiving of statistics.

As shown in figures 1 and 3, structure according to Galton nail-plate model, design positive six prismatic monomer structures of coupling normal distribution demonstrator: two bottom surfaces that highly are slightly larger than positive six prisms 5 of little bulb diameter vertically are bonded on two substrates 12 in front and back, utilize positive six prismatic top ribs as " nail ", the seamed edge of positive six prisms and the verticality of substrate are guaranteed by the technological level of machining; Distance between adjacent positive six prismatic 5 adjacent surfaces is slightly larger than little bulb diameter, equates with the height of prism, and it is foursquare pipeline that two adjacent surfaces and front-back baseboard 12 form cross sections; Pipeline has three kinds of directions: downwards, left, to the right, length is equal to the bottom surface length of side of positive six prisms 5; Positive six prisms 5 of following one deck are positioned at the centre of this pipeline, with this structure directing bead as much as possible along the axial-movement of downward pipeline, be bordering on and vertically remove to collide down one deck " nail ", positive six prismatic 5 monomers are arranged according to diagramatic way forms positive six prism honeycomb array 2 again; As shown in Figure 1, from horizontal direction, positive six prisms 5 of positive six prism honeycomb array on the edge, all the other each all be in the centre position of the last layer vertical pipeline relative with it with following one deck; Meet " nail " rib of certain positive six prism 5 when bead after, it only can be rolled in the pipeline on this rib left side or the right, and the probability of two pipelines equates about entering, and all is 1/2; Realize that the guiding bead moves towards normal distribution.Duct length in positive six prism honeycomb array is set to 2 times of little bulb diameters, the impulsive force that bead is vertically fallen is unlikely excessive and " nail " rib that align six prisms 5 damages, influence experimental precision, the positive six prismatic 5 bottom surface length of sides that adopt in this demonstrator approximately are 2 times of little bulb diameters, and the operation of assurance bead is unobstructed can not to be stuck in the pipeline.

As shown in Figure 1, the present invention is designed to two couplings between the normal distribution model with positive six prism honeycomb array, two normal distribution model are independent separately before not being coupled, be independent of each other, the pipeline of two coupling bodies begins to intersect at the m layer, promptly the first coupling body rightmost pipeline and the second coupling body Far Left pipeline combine together, make two groups of beads to enter in the pipeline of the other side institute " administration " by coupling layer, form coupling; The ratio that the number of plies that pipeline intersects accounts for total number of plies is more little, and it is loose more to be coupled, and coupling effect is not obvious more; Ratio is big more, and it is tight more to be coupled, and coupling effect is obvious more, even the overcoupling situation that two formation that distribute have only a peak can occur, as shown in Figure 6.

As shown in Figure 2, arrange four layers " nail " in going into ball device 1: ground floor and the 3rd layer are respectively positive six prisms 5, and the second layer left side still is positive six prisms 5, the right special-shaped block 1-1 that arranges, and the 4th layer is removable positive six prismatic 1-2.Bead is cast inlet by gentle slope 1-3 impact reducing,, make vertical drop as far as possible when bead has just begun to enter pipeline at vertical baffle 1-4 of inlet the right design.Bead bumps ground floor " nail " and enters left and right sides pipeline probability afterwards and be 1/2, begin the left and right sides from the second layer different structures is arranged, the left side still is positive six prisms 5, a special-shaped block 1-1 arranges on the right, to form different goal ratios: suppose that the bead that enters adds up to S in the 1. road of Fig. 3,2. the road and 3. the little nodule number in road then be S/2,4. the road and 5. the bead number average in road be S/4, through the 3rd layer of collision, with the bead mean allocation in 5. road to 7. road and 8. road, the 4th layer positive six prisms are removable, can directly cause the bead number of going into the interruption-forming different proportion of two coupling bodies of the right and left to distribute: in the time of in the middle of removable positive six prismatic 1-2 are positioned at, 8. the bead mean allocation in road arrives the left and right sides, finally forms left and right sides ratio and is

When shifting to the left side, 8. the bead in road all enters the right side, finally forms left and right sides ratio to be

When shifting to the right, 8. the bead in road all enters the left side, and the final left and right sides ratio that forms is

Like this, the present invention can form three kinds of different proportions and score total by going into particularly removable positive six prisms of ball device design polymorphic structure.

The design of removable positive six prisms as shown in Figure 4, open a mouth that is slightly larger than positive six prismatic 5 floorages at the correspondence position of metacoxal plate 12, the size of opening decides with the upper limit of positive six prismatic move left and right distances, getting the move left and right upper distance limit in this demonstrator is 1/3 little bulb diameter, removable " nail " when taking back bead roll to the right, removable " nail " when taking over bead roll left.Removable positive six prismatic 1-2 are bonded on the horizontal bolt 6, and horizontal bolt 6 is longer than A/F, by the 7 fixing horizontal bolts 6 of two draw-in grooves about opening it can only laterally be moved, and horizontal bolt 6 moves and drives positive six prismatic 1-2 simultaneously and move; Horizontal fastening when two scales overlap with edge of opening on 6 represents that positive six prismatic 1-2 move to open centre, and move left and right can move on to the edge of opening, reaches three kinds of states; Horizontal bolt 6 is fixed by the screw 8 above the draw-in groove 7, is fixed behind mobile the finishing and tests.

As shown in Figure 5, storage ball box 4 is designed to rectangular box.Pipe 3 is corresponding one by one with storage ball box 4 owing to receive, so its width is limited by the distance between duct width and the pipeline, length and highly can designing voluntarily receives can hold this that to manage 3 interior whole beads be criterion; The front of storage ball box 4 settles a handle 10 to be convenient to pull, and a square mouthfuls 9 is opened in the upper surface rear end, will store up ball box 4 during test and push the rear to, is tightly being supported to receive by opening upper surface portion not and is managing 3 opening, makes bead heap-shaped in the pipe of receiving become goalpost.After experiment finishes, will store up ball box 4 and extract certain distance out, and make the opening 9 of upper surface aim at the pipe of receiving, bead falls in the storage ball box 4 from opening part automatically in the pipe, and the distance of twitch is limited by draw-in groove on the base plate, and all storage ball boxes 4 adopt same size.

As shown in Figure 1, consider coupling desired stability of normal distribution demonstrator and workability, the present invention chooses organic glass and has made a coupling normal distribution demonstrator as material, and choose steel ball and experimentize, prebasal plate adopts transparent organic glass, metacoxal plate adopts blue organic glass, positive six prismatic 5 is blackish green organic glass, adopt these colour match both attractive in appearance, can observe bead again intuitively and fall to bumping stochastic process in " nail " process, and last bead distribution pattern.Utilize numerical control method to process positive six prisms 5, guaranteed that positive six prisms are bonded in the verticality of front-back baseboard, and the relative position of each six prism meets the demands and is not difficult to realize.Positive six prismatic total numbers of plies are many more, and the demonstration effect is approached normal distribution more.But be limited to structure size and cost, total number of plies of demonstrator of the present invention is elected 36 layers as, begins coupling by the 13rd layer, and the distribution situation under three kinds of ratios can be observed bimodal necessary being shown in Fig. 7 (A), Fig. 7 (B) and Fig. 7 (C).

As shown in Figure 9, for stability and the aesthetic property that increases this model, whole model is supported by stainless steel frame 11 and stainless steel diagonal brace, fills out sand in the steelframe and is used for counterweight.

Each selection of parameter of coupling normal distribution demonstrator of the present invention is: the regular hexagon length of side of positive six prismatic 5 bottom surfaces is 13mm, and height is 8mm, and the distance between the adjacent surface of adjacent prism is 8mm, i.e. duct width, the bottom pipe 3 width 8mm that receive, and length is 750mm; Choose storage ball box 4 and be rectangular parallelepiped box 65 * 80 * 25mm; The sphere diameter of steel ball is 6.35mm, and stainless steel stent 11 is 38 * 38 square standpipe, and stainless steel diagonal brace specification is 38 * 18.

Emulation experiment

According to notional result, estimate for the confirmatory experiment result and to test and to be convenient to design the volume of pipe 3 length and storage ball box 4 of receiving, carried out numerical simulation; Get bead and add up to 1000 pieces, went into ball device 1 formed three kinds of different proportions 1: 1 by aforementioned, 3: 5, the bead of two coupling bodies add up to about can obtain entering at 7: 9: 500: 500, and 375: 625,438: 562; Total number of plies of this demonstrator is 36 layers, begins coupling by the 13rd layer, can obtain following tables of little nodule number in following 49 pipes 3 of receiving of three kinds of ratios according to aforementioned relational expression: (the integral body numbering of the pipe 3 of receiving is shown in Fig. 1 below)

Table 1500: the distribution under 500 situations

The ball receiving groove numbering	1	2	3	4	5	6	7
								Little nodule number	0.0000	0.0000	0.0000	0.0001	0.0004	0.0027	0.0142
The ball receiving groove numbering	8	9	10	11	12	13	14
								Little nodule number	0.0607	0.2202	0.6850	1.8495	4.3714	9.1072	16.8132
The ball receiving groove numbering	15	16	17	18	19	20	21

Little nodule number	27.6217	40.5119	53.1722	62.5578	66.0445	62.6158	53.3919
								The ball receiving groove numbering	22	23	24	25	26	27	28
Little nodule number	41.1968	29.4712	21.1846	18.2143	21.1846	29.4712	41.1968
								The ball receiving groove numbering	29	30	31	32	33	34	35
Little nodule number	53.3919	62.6158	66.0445	62.5578	53.1722	40.5119	27.6217
								The ball receiving groove numbering	36	37	38	39	40	41	42
Little nodule number	16.8132	9.1072	4.3714	1.8495	0.6850	0.2202	0.0607
								The ball receiving groove numbering	43	44	45	46	47	48	49
Little nodule number	0.0142	0.0027	0.0004	0.0001	0.0000	0.0000	0.0000

Distribution under 375: 625 situations of table 2

The ball receiving groove numbering	1	2	3	4	5	6	7
								Little nodule number	0.0000	0.0000	0.0000	0.0000	0.0003	0.0021	0.0106
The ball receiving groove numbering	8	9	10	11	12	13	14
								Little nodule number	0.0456	0.1651	0.5137	1.3871	3.2786	6.8304	12.6099
The ball receiving groove numbering	15	16	17	18	19	20	21
								Little nodule number	20.7163	30.3839	39.8794	46.9197	49.5404	46.9922	40.1540
The ball receiving groove numbering	22	23	24	25	26	27	28
								Little nodule number	31.2401	23.0281	18.0742	18.2143	24.2951	35.9142	51.1535
The ball receiving groove numbering	29	30	31	32	33	34	35
								Little nodule number	66.6298	78.2393	82.5485	78.1958	66.4650	50.6398	34.5271
The ball receiving groove numbering	36	37	38	39	40	41	42
								Little nodule number	21.0165	11.3839	5.4643	2.3118	0.8562	0.2752	0.0759
The ball receiving groove numbering	43	44	45	46	47	48	49

Little nodule number

0.0177

0.0034

0.0005

0.0001

0.0000

0.0000

0.0000

Distribution under 438: 562 situations of table 3

The ball receiving groove numbering	1	2	3	4	5	6	7
								Little nodule number	0.0000	0.0000	0.0000	0.0000	0.0004	0.0024	0.0124
The ball receiving groove numbering	8	9	10	11	12	13	14
								Little nodule number	0.0532	0.1929	0.6000	1.6201	3.8294	7.9779	14.7284
The ball receiving groove numbering	15	16	17	18	19	20	21
								Little nodule number	24.1966	35.4884	46.5790	54.8013	57.8585	54.8665	46.8259
The ball receiving groove numbering	22	23	24	25	26	27	28
								Little nodule number	36.2583	26.2754	19.6419	18.2143	22.7274	32.6669	46.1353
The ball receiving groove numbering	29	30	31	32	33	34	35
								Little nodule number	59.9579	70.3650	74.2305	70.3142	59.7654	45.5353	31.0468
The ball receiving groove numbering	36	37	38	39	40	41	42
								Little nodule number	18.8980	10.2364	4.9135	2.0788	0.7699	0.2475	0.0683
The ball receiving groove numbering	43	44	45	46	47	48	49
								Little nodule number	0.0159	0.0031	0.0005	0.0001	0.0000	0.0000	0.0000

Manage the 3 distribution histograms of numbering with receiving shown in Fig. 7 (A) according to the bead number that table 1 obtains.Manage the 3 distribution histograms of numbering with receiving shown in Fig. 7 (B) according to the bead number that table 2 obtains.Manage the 3 distribution histograms of numbering with receiving shown in Fig. 7 (C) according to the bead number that table 3 obtains.Can see coupling between the normal distribution intuitively by histogram, also can determine in total number of plies to be that coupling layer is chosen to be 13 under 36 the condition, can clearly see three kinds of coupling normal distribution general pictures under the ratio.

Effect of the present invention can further specify by test findings.

Regulate removable positive six prismatic 1-2, be fixed, 1000 pieces of steel balls are dropped in the inlet of top along gentle slope 1-3 successively, add up the little nodule number in the pipe 3 of respectively receiving then, the statistical value of normal distribution obtains being coupled, and compare with the theoretical value of coupling normal distribution, find that the statistical value and the theoretical value of coupling normal distribution has very high correlativity, this has also illustrated the correctness of this demonstrator.

Claims (8)

1. coupled normal distribution probability demonstration instrument, on the substrate of demonstrator, arrange n layer nail, nail equates with distance between the nail, each nail all is in the centre position in last layer or the hole that one deck is relative with it down, constitute bead operation passage, after bead is met certain nail, the probability of two passages equated about it entered, it is characterized in that: described nail is made of positive six prisms (5), several positive six prisms (5) vertically are bonded in front and back two substrates (12), the space that the adjacent surface of every adjacent two positive six prisms and front-back baseboard (12) thereof form constitutes bead operation " pipeline ", with with over against the seamed edge of positive six prisms of bead traffic direction as " nail ", following one deck positive six prismatic those ribs as " nail " are relative with the pipeline bottom center of last layer, guarantee bead bumped nail and fell into about the probability of two pipelines equate and fall along pipeline, vertically remove to bump down " nail " rib of positive six prisms of one deck, the guiding bead moves towards normal distribution; Described substrate (12) is provided with into ball device (1), positive six prism honeycomb array (2), receive pipe (3) and storage ball box (4), describedly go into ball device (1) and be provided with positive six prisms (5), special-shaped block (1-1) and removable positive six prisms (1-2) by the binomial distribution principle, be used to make up the different goal ratio of two-way bead, as after the basis that is coupled between the different number colony; Go into the left and right arms goal passage of ball device (1) by symmetry and connect positive six prism honeycomb array (2), described positive six prism honeycomb array by two independently positive six prisms " nail " model of normal distribution be coupled to form, last one deck " nail " below pipeline of positive six prism honeycomb array (2) is made as long pipeline, as the pipe (3) of receiving, the bead that storage receives, the equal in length of all pipe (3) of receiving, arrange front of formation, the bead post in each pipe of front constitutes the bead coupled normal distribution probability figure that intuitively shows by positive six prism honeycomb array; In the below of each pipe (3) of receiving a storage ball box (4) is set, is used for cooperating the bead number that forms in receive pipe goalpost and each pipe (3) of receiving of statistics.

2. coupled normal distribution probability demonstration instrument according to claim 1, it is characterized in that: describedly go into to arrange four layers of nail in the ball device (1): ground floor and the 3rd layer are respectively positive six prisms (5), the second layer left side still is positive six prisms (5), the right special-shaped block (1-1) of arranging, the 4th layer is removable positive six prisms (1-2), probability is 1/2 to the left and right sides after bead bumps ground floor " nail ", begin the left and right sides from the second layer and form different ratios, through the 3rd layer of removable positive six prism (1-2) that collides the 4th layer, directly cause the bead number of going into three kinds of different proportions of interruption-forming of two coupling bodies of the right and left to distribute.

3. according to the coupled normal distribution probability demonstration instrument of claim 1 or 2, it is characterized in that: described removable positive six prisms (1-2) are bonded on the horizontal bolt (6), removable positive six prisms (1-2) correspondence position of going into ball device (1) at metacoxal plate is opened a mouth, this opening size is by the upper limit decision of positive six prismatic move left and right distances, horizontal bolt (6) is fixed in the draw-in groove (7) of opening both sides it can only laterally be moved, horizontal bolt (6) moves and drives positive six prismatic moving simultaneously, two scales are arranged on horizontal the fastening (6), be used for controlling the shift position, removable positive six prisms (1-2) move to open centre and left and right edges, form three kinds of states of scoring.

4. coupled normal distribution probability demonstration instrument according to claim 1, it is characterized in that: bead two independently the m layer pipeline of positive six prism honeycomb array of normal distribution (2) begin to intersect, promptly the first coupling body rightmost pipeline and the second coupling body Far Left pipeline combine together, making two groups of beads enter by coupling layer in the pipeline of the other side institute " administration " is coupled, if the coupling number of plies is m, have (2m-1) individual pipe (3) of receiving to merge mutually, final pipe (3) number of always receiving that forms is (2n-2m+1).

5. coupled normal distribution probability demonstration instrument according to claim 1 is characterized in that: according to binomial distribution, derive receive after two coupling bodies interact pipe (3) total in the number of bead with the relation of the number of plies n and the cambium layer m that is coupled:

Wherein, S is the bead sum, and the bead number that enters first coupling body is pS, (0＜p＜1), and the little nodule number that enters second coupling body then is (1-p) S, l ₁Be the symmetrical distribution numbering of first coupling body, span is | l ₁|≤[(n+1)/2]; l ₂Be the symmetrical distribution numbering of second coupling body, span is | l ₂|≤[is (n+1)/2], wherein [(n+1)/2] represents the smallest positive integral bigger than (n+1)/2, n is odd number l ₁And l ₂All get odd number, n is even number l ₁And l ₂All get even number; Satisfy condition | l ₁+ l ₂| the pipeline of≤2 (m-1) is the fusion pipeline after being coupled; Constitute coupling normal distribution figure by the bead post in each pipe (3) of receiving.

6. according to the coupled normal distribution probability demonstration instrument of claim 1, it is characterized in that: on a design gentle slope (1-3), one side of going into ball device (1) porch, buffering bead whereabouts dynamics; At an another side design vertical baffle (1-4), make vertical drop as far as possible when bead has just begun to enter pipeline.

7. coupled normal distribution probability demonstration instrument according to claim 1 is characterized in that: positive six prisms (5) the bottom surface length of side is 2 times of little bulb diameter, and the impulsive force of avoiding bead vertically to fall is excessive and be stuck in the pipeline.

8. coupled normal distribution probability demonstration instrument according to claim 1, it is characterized in that: the structure of described storage ball box (4) is rectangular box, its width is limited by the distance between duct width and the pipeline, the positive handle (10) of settling of box body is convenient to pull, the distance of twitching is opened a square orifice by the restriction of the draw-in groove on base plate cartridge upper surface rear end, be used for collecting pipe (3) bead of receiving after experiment finishes, by the pull box body, square orifice on the box body is communicated with the pipe (3) of receiving or misplaces, bead coupling demonstration or collection in the pipe (3) of realizing receiving.

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