IntroductionA more general question would be: "What is the probability of the number of points 1~6 appearing (x1,x2,x3,x4,x5,x6)?" where sum (x1~x6) = n ". This is a polynomial distribution.definitionMultiple distributions are obtained by extending the two distributions to multiple (greater than 2) mutually exclusive event occurrences. A typical example of a two-item distribution is the toss of a coin, the probability of the coin facing up to p, the repetition of the n-th coin, and the probability of the positive of the K-th as a two-item distribution probability. (Strictly defined in the two-item distribution in the "Burlington" Experimental definition) to expand the two to a number of items to obtain a number of distributions. such as throwing dice, different from tossing a coin, the dice have 6 sides corresponding to 6 different points, so that the probability of a single point facing up is 1/6 (corresponding to P1~P6, their value is not necessarily 1/6, as long as and for 1 and mutually exclusive, such as a shape of irregular dice), repeatedly throw n times, If you ask for X times is the number of points 6 upwards of the probability is:. A more general question would be: "What is the probability of the number of points 1~6 appearing (x1,x2,x3,x4,x5,x6)?" where sum (x1~x6) = n ". This is a polynomial distribution problem. Then simply use the formula above the idea of the multiplicative reduction will get the following Figure 1 probability formula. A random experiment if there are k possible endings A1, A2 、...、 Ak, respectively, their occurrences are recorded as random variables X1, X2 、...、 Xk, their probability distribution is P1,P2,...,PK, then in the total results of n-sampling, A1 appears N1 times A2 appears N2 times 、...、 The probability that the occurrence of this occurrence of the NK occurrence of AK has the following formula: write in another form:2 Formula application editing probability formulaThis is the probability formula for multiple distributions. It is obvious that it is a polynomial distribution because it is a special type of polynomial expansion. We know that in algebra, when K variables and the N-square expansion (p1+ p2+...+ PK) ^n is a polynomial, the general term is the value given by the previous formula. If this k-variable happens to be the probability of the occurrence of various outcomes, then the probability of the total value of these probabilities corresponds to an inevitable event. and the probability of the inevitable event is equal to 1, so the above polynomial becomes (p1+ p2+...+ PK) ^n =1^n=1, that is, the value of the polynomial is equal to 1. Because (p1+ p2+...+ PK) The value of ^n is equal to 1, we also think that it represents an inevitable event of the probability of N-sampling (=1, inevitable event). And when the polynomial can be expanded into many items, the total value of these items equals 1 suggests that we are the corresponding probabilities of some incompatible events (n-sampling), that is, each item of a polynomial expansion is the probability of occurrence of a particular event. So we put the expansion of the general term as A1 appear N1 times, A2 appear n2 times, ..., AK occurrence of this event of NK occurrence probability. This gives you the previous formula. If the probability of occurrence of individual events p1,p2,...,pk equal, i.e. p1=p2=...=pk=p (note here is lowercase p), note that p1+p2+...+pk = 1, you get p1= p2 =...=pk =p=1/k. By substituting this value for the expansion of the polynomial, the sum of the individual items of the expansion satisfies the following formula: ∑[n!/(n1!n2!... nk!)] (1/k) ^n=1 namely ∑[n!/(N1!n2!... nk!)] =k^n The sum above all the possible positive integer values of each NI, but requires that the aggregate value of each NI equals N. That is n1+n2+...nk=n.ApplicationA situation that is used to process multiple possible results of an experiment. When thermodynamics discusses the possible number of microscopic states of matter, it often leads to n!/(n1!n2!... nk!) with additional ideas. Expression and called it a thermodynamic probability. It is a much larger number than astronomical, and it is not appropriate to call it probability. But in thermodynamics the probability of the occurrence of each microscopic state is equal, which corresponds to the p1= p2 =...=pk =p=1/k We discussed earlier, so [n!/(N1!n2!... nk!)] (1/KN) really has the meaning of mathematical probability. In other words, the thermodynamic odds in physics [n!/(N1!n2!... nk!)] The probability of (having a normalization) defined in mathematics after multiplying (1/KN).
Multi-item distribution (multinominal distribution)
