Stochastic event probability gambling poisson distribution

Probability theory is a branch of mathematics that studies the law of random phenomena. Its origins in the 17th century century, at that time in the category of error, demographic, life insurance, need to collate and study a large number of random data, which bred a special study of a large number of random phenomena of the regularity of mathematics, but at that time to stimulate mathematicians to think first of the problem of probability theory, but it is from the gamblers problem. Mathematician Fermat to a French mathematician Pascal asked the following questions: "The existing two gamblers to bet on a number of innings, who won the first S Bureau, when the Gambler won a [a < S], while the Gambler B won B [B < s], gambling stop, that bookie should be how to divide only reasonable? "So they set out on different grounds and gave the right solution on July 29, 1654, and three years later, in 1657, another Dutch mathematician, Whigens [1629-1695], solved the problem in its own way, and wrote a book on the calculation of gambling, This is the first theory of probability theory, the three of them put forward the solution, all involved in the mathematical expectation [mathematical expectation] this concept, and thus lay the foundation of classical probability theory. The other founder of   making probability theory a branch of mathematics is the Swiss mathematician Jacob-Bernoulli [1654-1705]. His main contribution is to establish the first limit theorem in probability theory, which we call the "Bernoulli large number theorem", that is, "in repeated experiments, the frequency of the more stable trend." This theorem was even more in his death, 1713, published in his posthumous "guessing."   By the year 1730, the French mathematician, Moivre, published his book, "Analysis Miscellaneous", which contained the famous "Di-Laplace theorem". This is the primitive initial form of the second basic limit theorem in probability theory. And then Laplace in 1812 published the "Theory of Probability Analysis", the first explicitly to the probability of a classical definition. In addition, he and a number of mathematicians established the "normal distribution" and "least squares" theory. Another representative figure in the history of probability theory is the Poisson of France. He promoted the law of large numbers under Bernoulli form, and a new distribution, the Poisson distribution, was obtained. The theory of probability after them, the Central research project focuses on the promotion and improvement of the law of large numbers and the central limit theorem.   probability theory developed into the 1901, the central limit theorem was finally rigorously proven, and the latter was used by the mathematician for the first time to scientifically explain why many random variables encountered in practice are approximated to the normal distribution. In the 20th century's 30 's, people began to study the stochastic process, and the famous Markov process theory in 1931 years before it was laid its place. The Soviet mathematician Kolmogorov in the history of probability theory also made a significant contribution to the modern times, the emergence of theoretical probability and application of the probability of the branch, and the use of probability theory to different categories, thusDifferent disciplines have been carried out. Therefore, modern probability theory has become a very large branch of mathematics.   gambling vs. Random events   The right attitude towards gambling has two kinds. If you really want to think of gambling as entertainment, the loss of money as a cost to pay for this, then, should be in advance to consider the entertainment value of how much money, so that the "consumption" strictly limited to this number within. If you really want to win the casino, you have to spend enough time to understand and master the knowledge of gambling. There are two kinds of phenomena in the nature of  , one is called the decisive phenomenon, and the characteristics of this kind of phenomenon are: under a set of conditions, the result is completely decided, either fully affirmed or totally negative, there is no other possibility. The decisive phenomenon is actually the phenomenon that can predict the result beforehand.   There is also a class of phenomena called non-deterministic phenomena characterized by conditions that do not fully determine the outcome, and each time the result may be different. The non-decisive phenomenon is actually the phenomenon that can not predict the result beforehand, only then can know exactly what the result of it happens, in probability theory, this kind of phenomenon is called stochastic phenomenon.   at any one time in a randomized trial, the results are not accurately predicted before the experiment, which is a no-proof conclusion in probability theory, and as a precise mathematical discipline, probability theory studies the regularity of a large number of random trials. Take the roulette, each time the roulette out what number is not accurate prediction-this is the basic function of the roulette wheel, but in countless experiments or the number of experiments enough, roulette delineated is completely regular, from a large number of roulette delineated data and many people's roulette practice can be found long bet will lose, No bet is to win this roulette truth.   Gambling is a random phenomenon refers to the gambling every win and loss is irrelevant to the prediction, regardless of who guessed, its guessing probability and guess the person has nothing to do, is a constant, so the casino never guessed, and the vast majority of gamblers but endless guess to guess. In fact, the people who like gambling are very smart, are very hard, but the biggest mistake of ordinary gamblers is that the casino provides a record of the number of roulette, can be found in the delineated data every time the law of roulette delineated, and it in turn to guide the prediction of the ball will fall to which number or which area Think that in this interaction process constantly revised to improve technology, there is always a day to win the casino. Ordinary gamblers because of the guiding ideology and research method is not correct, the conclusion naturally is absurd, but thought that the loss of money is due to their own technology, and thus more qinxuekulian, hoping to achieve the goal of the day, unknowingly caught in the more gambling lost, the more lost gambling circle, this is an endless vicious circle. Casino for ordinary gamblers prepared roulette record paper and baccarat record paper, not because the casino there is how noble, it is misleading gamblers, let you enter the vicious circle, self-control and the strong may be less with or simply not with the casino, a few people may therefore go astray, suffering from pathological gambling disease.   Gambling is not only a random experiment, but also a classical model test, so the probability of gambling can be accurately calculated,Just some simple, almost no need to think about, some complex, must rely on the use of computers and ingenious algorithms. For example, the number "0", "1", "2" appears in roulette. Until the "36" and so are the basic events, and the size, red and black, single-pair is composed of basic events composite events; Pull the cell, any five cards are the basic events, a total of 2598960, and pairs, double batches, three bar ... All the way to the same Huatashun is a composite event composed of basic events; 21 points of the situation is more complex, the dealer from the card box each issued a card is the basic event, and appears "2", "3", "4" ... Until "K", "a" card is a composite event (because each card has four suits); Similarly, the dealer from the card box has taken out two cards is also the basic event, and these two cards are composite events; Generally, taking a number of cards in turn from the card box is a basic event, and the points of these cards are composite events. In all the play, losing or winning is a very complex compound event.   Each game has a lot of random variables, some of which are unique. For example, the face value of the next card in 21 points is a random variable, which can be taken from 1 to 11 of any one integer; The dealer is a regular card, and the card is also a random variable, it can be a value from "17" to "21" Any integer, also includes "Blackjack" and " Two pips; For example, the face value of the next card in baccarat is also a random variable, it can be taken from 0 to 9 of any one integer; Village idle points are also a random variable, the value can be from "0" to "9" any integer between.   No matter what the gambling play, are to lose or win as a result of gambling, lose and win are random events, to digitize them, in which, the loss of negative, win as a positive number, the value with the change in the gambling results of a random variable-odds, which is the most important in gambling a random variable, is necessary for any kind of gambling play.   Gambling as a random experiment, probability analysis is the effective way to study gambling, it involves some preliminary knowledge of probability theory and modern computing means, as long as not gambling God, its gambling will inevitably obey by a variety of probabilities determined by the outcome of the relationship, The key to winning the casino is to see if there is a good chance of betting on the gamblers.   Probability and PredictionThe Ancients cloud: All things pre-set, not pre-waste, stressed that no matter what to do in advance planning, pre-design, which can not be separated from the law of things and phenomena of understanding. For deterministic phenomena, only a clear causal relationship can predict the result accurately. But to the stochastic phenomenon, as long as know the probability can be predicted, but it should be noted that the probability to predict is not the result of random events, but a large number of random events results in the quantitative regularity. For example, if you throw a coin, you cannot say whether it is positive or negative, you can only say: "There are one-second positive opportunities", and if someone says, "There are one-third positive opportunities", no matter what side it is, these two conclusions can not be reflected But if it is thrown 100 times or more, such as 10,000 times, then the "One-third chance to be positive" argument is obviously untenable, and the "one-second chance to be positive" statement can be reflected to a considerable extent. Here we describe in detail the principle of probability prediction. Law of large numbers  Under the same conditions for a large number of experiments, according to the frequency stability, the frequency of event a must be stable near a certain constant P, then the probability of defining event A is:  p (a) =p  this is called the statistical definition of event probability, the corresponding probability is called statistical probability, The statistical definition of probability gives an approximate method of calculating event probability, that is, when the number of experiments is sufficiently large, the frequency of the available events is the approximate value of the probability of the event. It is not understood, however, that the more times the test is, the more likely the event is to be the probability of the event. For example, in the case of a coin toss, a person throws two times, a positive one on the opposite side, a positive frequency of 0.5, exactly equal to a positive probability, while the other person does the same experiment, throws 10,000 times, 4,985 times the positive, and the positive frequency is 0.4985, instead of Equal to the probability of a positive, this throw 10,000 times than throw two times the result of high precision, then this more than 9,998 times is not white throw it? To explain this phenomenon, the relationship between frequency and probability must be studied in more detail.   In fact, the frequency is a random variable, and there are many or even countless possible values, which can be any number between 0-1. The probability is a fixed constant, which is a definite number between 0-1. We are interested in a region of probability, which may fall in or out of the region; for the number of test times n, the frequency falls within the region of the event also has a probability, when the number of tests n increases, this probability also increases; when the number of trials increases infinitely, the area becomes infinitely smaller , the probability that the frequency falls within the region will be equal to 1.   Generally, the relationship between frequency and probability is not expressed in the ordinary equation, but in the probability that the frequency and probability of the event falls within a certain range, namely:  p (|μn/n―p|<ε)   The size of the specified ε, This probability can be calculated by using knowledge of the probability theory that concerns the inequalities of the snow.   When the number of tests n infinitely increases the conclusion is that the law of large numbers. The law of large numbers is the general term of a series of laws in probability theory, also known as "large number law" or "average law", is one of the main laws of probability theory.   Historically, Bernoulli first proposed a law of large numbers. In layman's terms, the theorem is that, under the condition of constant experimentation, repeated experiments have several times, and the frequency of random events approximates its probability.   In addition to textual representations, the law of large numbers also has a precise mathematical representation.   In the Bernoulli test, when the number of times n is infinitely increased, the frequency of event a μn/n (μn is the number of occurrences of event A in n trials), which converges to its probability p. That is, for any ε> 0, there are:  lim P (|μn/n―p | <ε) = 1n→∞  This is the law of Bernoulli large numbers. Of course, the above formula looks a little laborious, it does not matter, because everyone understands its text expression, in fact, for gamblers, the large number of laws of the text expression has more realistic guiding significance. The statistical definition of probability "frequency stability in probability" meaning is very unclear, Bernoulli large number theorem from the mathematical clarity of the problem, "frequency stability in probability" meaning is: the frequency of event a μn/n according to the probability of convergence to its probability p, that is, when n full large can be any close to 1 probability assertion, μn/ N will fall in the ε region centered on p.   The law of large numbers expresses the law of random experiment in definite mathematical form, and proves the condition of its establishment, and expounds the regularity of "frequency stability in probability" which is presented in a certain condition and repeated random phenomenon in theory. As a result of the law of large numbers, the whole effect of a large number of random factors will inevitably result in some kind of independent random events.   If probability theory is about stochastic phenomenon prediction theory, then the law of large numbers tells us the method of prediction and how to predict it. Bernoulli the law of large numbers from the theory of the test to determine the probability of the method: do n independent repetition test, to μn to indicate n test a occurrence of the number of times, when N is large enough, then we can be very large probability of certainty: p≈μn/n. We often use this method when the probability of an event is unknown or if the probability of the calculation of the theory is accurate.   in turn, the probability of a known event, when N is large enough, can be used to predict the number of occurrences of an n-Bernoulli test with the probability of an event: ΜN≈PXN, where n is larger, the prediction is more credible. Any gambling play in the casino every time only win and not win two kinds of results ("and" or "ping" can be regarded as 50% win), gambling is Bernoulli test. Accurate calculation of the game's win rate, can be used to predict the outcome of gambling, the basis is the law of large numbers. The longer the bet, the more effective the prediction will be.   can now explain the aforementioned phenomena. Throw two coins, it is possible to appear twice is positive or two negative situation, the frequency as the probability is obviously wrong, that is, the frequency of throwing two coins as a probability, the probability of serious deviations up to 50%, And the frequency of tossing 10,000 of coins as a probability in most cases results are quite credible. The conclusion is that the test 10,000 times more credible than the results obtained two times, does not violate the intuition told us.   Therefore, using statistical methods to determine the probability of an event, the frequency with the increase in the number of trials close to the probability is also in the way of probability. The more the number of statistics, the more likely the probability of the frequency approximation, the more credible the results, it can be thought that the statistics reflect the credibility of the results, and at this time the frequency of the result and probability of how close to a certain randomness. In other words, by experimenting to determine the probability that a windRisk, in any case, there is a frequency deviation from the probability of the situation exists, increase the number of trials, can reduce the risk, but not to eliminate the risk itself, only in the case of infinite number of trials, there is no such risks. However, when the number of trials is long enough, although the frequency as a probability or the possibility of error, but this possibility is very small, so that can be completely assured without fear of error.   gambling is the probability of betting   Roulette 10 times the red, some people feel that the 11th time the black, even out 20 times red, the 21st time should be more black ... As a result, in the gambling often encountered in the successive large after the small, successive out of the village after the bet idle, even after the loss of filling and other wrong methods, known as the reverse gambling method, the reverse betting method with the changes in the bet has been widely popular in the casino "injection code Method", and there seems to be a more sufficient reason: in multiple consecutive betting, as long as the win once, will be able to win back all the previous losses, and win a little more, it is necessary to clarify it.   This kind of inverse gambling method has a feature, is that the probability has been known and close to One-second, for example, we can say that the probability of throwing a coin out of the positive is 1/2; roulette on the other side, except 0, the number of red and black is equal, no doubt the probability of red and black out is equal and close to One-second ... This gives us a sense that the probability of a random event can be shown at any time as a property. In the stock market, the probability of the rise and fall is unclear, so we all chase up the fall, but also few people use the method of injection code, the performance of the exact opposite.   For a long time, people are accustomed to the rule of defining relationships with only one result, for example, in times when a festival is getting closer, and we even use a countdown to express this relationship; at a distance, as soon as we move towards the destination, we will be closer to it, and we are accustomed to this physical approach , which is usually getting closer. But still not accustomed to the aloof, the overall situation is approaching the probability of the approach, the probability approach means that some time away from near, some time away from far, not close to is very natural, for example, in small samples, the frequency will occasionally focus on the probability, in large samples, the frequency will be concentrated in the probability near, However, whether large or small sample, can not avoid the frequency of serious deviation from the probability of such a situation, and people are accustomed to the rule of law from the absence of exceptions, for the small sample often continuous red such a serious deviation of the situation is an anomaly, in the subsequent trials will soon be corrected; in fact, Roulette has no memory, Remembering previous results and correcting them is not roulette. The substitution of deterministic relationships for probabilistic relationships between objects is a mistake that people unknowingly make.   frequency and probability of the relationship between the probability to describe, usually the two are unequal relations, generally can not be equated, only when the number of trials is very large, only μn/n≈p, and there is always the possibility of an exception error. recognize the frequency and probability of this relationship, will help to overcome the continuous large after the small, successive out of the village after the bet idle, even after the improper gambling psychology, such errors, the root of this kind of error is that the frequency and probability are not conditionally linked together with the equal.   Subconsciously, we have a prediction that the equal chance of throwing coins, the number of positive and negative times in successive tests should be very close,The relationship between frequency and probability indicates that there are often many times when this prediction is not allowed. Roulette out of 10 results, most of the time these 10 results in the proportion of red and black relatively close, if the 10 times red, only to indicate that the forecast is not allowed, like the weather forecast, if the forecast is not allowed for ten consecutive days, then the 11th day forecast will be more accurate? The average person would not think so, and we have more reason to think that there is something wrong inside the meteorological department and that the forecast will be more inaccurate. Of course, unlike the weather forecast, the prediction of roulette is not affected by human factors.   Worse than using probabilities to predict the frequency of a small number of trials, it is customary to use probabilities to predict the outcome of the next random event and to link it to the frequency of previous trials. In fact, no matter how far ahead the frequency and probability difference, continue to test, and then the frequency of the test is only related to the probability, and the previous frequency independent, and for the results of just one test, we can only say in general the probability of an event occurrence.   Probability is only used to predict the frequency of a large number of test reliability is very high, to improve the accuracy of prediction, only by improving the predicted range. If you predict the number of positives from the 11th to the 1010th times, you say that the positive is close to 500 times, which is much more accurate than predicting the 11th result.   from another point of view, large samples can be divided into a number of small samples, a small sample of a particular type of combination, such as continuous out of the positive as an event, this is a small probability event, by the law of large numbers is easy to deduce that in the long-term experiment, the small probability event is almost certain to occur, But people tend to think of it as a zero-probability event that does not appear and should not occur. In the coin toss test, the probability of the positive and negative is the same, is 50%, when the positive, will not produce the illusion of the opposite side immediately, the same test, when we do not continuously out "positive" and continuous "positive" as the object of observation, the probability of the two are very different, the former is much larger than the latter, Because the latter probability is very small, once appeared, immediately will produce this phenomenon should immediately terminate the illusion; in fact, the probability of continuous "positive" is small, is also a number not 0, as long as it is not equal to 0, as long as the test time is long enough, continuous "positive" will almost certainly occur, is an unavoidable phenomenon. Once there, it's as normal as throwing a coin out of the opposite side, no fuss.   Interestingly, it is also a small probability event, some we want it to happen, and some hope it doesn't happen. Gambling in the loss is gamblers do not want to happen, once it happens, always hope that this has happened to the small probability event can quickly terminate, so often in the continuous loss of the increased stakes. Another fact is that, for an individual, lottery is a small probability event, and we want it to happen to us, and if someone does not reject it because it is a minimal probability event, it will gladly accept the fact. Should be like accepting the lotteryTo accept the fact that there have been 10 consecutive red.   random trials and eventsThe stochastic phenomenon is characterized by a series of tests or observations that have different results in the same condition, and that the results will not be foreseen before the test or observation. Randomised trials: For experiments or observations of random phenomena, it must meet the following properties: (1) The possible results of each trial are not unique, (2) The results may not be determined before each trial, and (3) the test may be repeated under the same conditions. Random Events (events): In randomized trials, results may or may not occur. The result of the experiment may be a simple event or a complex event. Simple events are events that can no longer be decomposed, also known as basic events. Complex events are events that are composed of simple events. The basic event can also be called a sample point, and the experiment has n basic events, which are recorded as (i=1,2,..., N). Collections ω={ω1, ω2, ..., ωn} are called sample spaces, and the elements in Ω are sample points. Example: Throw a uniform hexahedral dice, there may be 1, 2, 3, 4, 5, 6 total six species. These six results are basic results and can no longer be decomposed into simpler results, so ω= {1,2,3,4,5,6} is the sample space for the experiment. The event "The number of points is odd" is not a simple event, it is a combination of the basic event {1},{3} and {5}. We usually use capital letters A,B,C, ... To indicate a random event, for example, set A to indicate "the number of points is odd", then a={1,3,5}; set B means "the number of points is even", then b= {2,4,6}. Definition of probability  probability refers to the probability of a random event, or probability, as a measure of the likelihood of a random event occurring. n Repeat test, the number of random event A occurs m times, the frequency is m/n, when the number of times n is large, if the frequency in a certain value of the oscillation, and as the number of times n increases, the frequency of the oscillation amplitude is more and more small, then the probability that P is the event A, recorded as: P (A) =p. In classical general situation, that is, the probability of the occurrence of the basic event is the same situation: P (a) =m/n  example: Set a bag with white Ball 2, black Ball 3. (1) Randomly touch 1 balls, ask what is the probability of a white ball?    (2) The probability of randomly touching 2 balls and asking 2 balls is a white ball. Q 2 What is the probability of a white and a black ball? Three questions 2 the probability of a black ball is how much?  solution: (1) The total number of sample points is n=5 because any 1 balls that are touched form a basic event. Use A to touch out is the white ball event, then A is composed of two basic points, namely a={white Ball, white ball}, favorable occasions number m=2. Therefore, the probability of just touching the white ball is P (a) =m/n=2/5=0.4 (2) as the 2 ball is a basic event, so the total sample point is 10 p (a) =p (2 balls are white ball) =1/10p (B) =p (2 ball one white one black) =2x3/10=6/10p (C) =p (2 balls are black balls) =3/10note:p (A+b+c) The basic properties   properties of =1  probability 1      1≥p (A) ≥0. Property 2      p (Ω) = 1. Property 3      If event A is incompatible with event B, that is Ab=ф, then P (a∪b) =p (A) +p (B). The probability of inference 1     Impossible event is 0, i.e. P (Ф) = 0. Corollary 2     p (a) =1-p (a),        represents the opposite of a, i.e. they must have an event but cannot occur simultaneously.   Probability Algorithm--addition formula for P (a∪b)--"A occurrence or B occurrence" probability mutex event (incompatible event) cannot occur simultaneously event without common sample point mutex event addition Formula P (A∪B) =p (A) +p (B) Complementary events cannot occur at the same time and there must be a probability that a two event complementary event will occur equal to 1  For example: Throw a dice, "2 points" probability is 1/6, then "No 2 points" probability is 5/6. Compatible events Two events may occur at the same time without common sample points compatible event addition formula (generalized addition formula) P (a∪b) =p (A) +p (B)-P (AB)   probability algorithm-multiplication formula is used to calculate the probability of simultaneous occurrence of two events. The probability that "a occurs and B occurs" is that P (AB) first focuses on whether the event is independent of the conditional probability-the probability of a conditional probability of a occurrence under some additional conditions that has occurred in a known event B--p (a| B) general formula for conditional probability: P (a| b) =p (AB)/P (b) The general form of the multiplication formula: P (AB) =p (A) · P (b| A) or    p (AB) =p (B) · P (a| B)   (1) Conditional probability p (a| b) = The probability of ab occurring in all possible results of B is the conditional probability P (a|) considered in the sample space Ω b), the probability of the event AB being calculated in the new sample space B the independence of the event two events the occurrence of an event does not affect the probability that another event occurs P (a| B) =p (A), or P (b| A) =p (B) multiplication formula for independent events: P (AB) =p (a) · P (B) extended to N independent events, with: P (A1 ... AN) =p (A1) P (A2) ... The arithmetic of probability of P (an)-full probability formula Complete Event Group Event A1, A2 、...、 an incompatible, a∪a2∪ ... ∪an=ω and P (Ai) > 0 (i=1, 2 、...、 N) for either event B, it always occurs simultaneously with the complete event group A1, A2 、...、 An, then there is a full probability formula for P (B): the algorithm of probability The visual meaning of the full probability formula of Bayesian formula: Every AI occurrence can cause B to occur, each AI causes B to occur probability, so the probability of event B as a result is the sum of the probabilities of each "cause" AI, in contrast to the observed condition that event B has occurred, Determine the probability of an AI that causes B to occur  --Bayesian formula (inverse probability formula)   (posterior probability formula) stochastic variable and its probabilistic distribution stochastic variable--the value of the variable that represents the result of the random test is random, It is not possible to determine in advance which value a value corresponds to a potential result of a randomized trial in uppercase letters such as x, Y, Z ... To indicate that the specific value is used in the corresponding lowercase letters such as x, Y, Z ... To show that according to the characteristics of the value of the different, can be divided into: discrete random variable--the value can be listed as a continuous random variable--the value cannot enumerate the probability distribution of discrete random variables x probability distribution--x finite possible valueCorresponds to the relationship between Xi and its probabilistic Pi (i=1,2,3,...,n). The probability distribution has the following two basic properties: (1) pi≥0,i=1,2,..., N; (2) Σpi=1 discrete probability distribution: probability function: P (x= xi) = Pi distribution column: The probability distribution of the probability density continuous random variable of the continuous type random variable of the distribution graph can only be expressed as: mathematical function-- Probability density function f (x) and distribution function f (x) Graph    --probability density curve and distribution function curve probability density function f (x) function value is not probability. The probability that a continuous random variable takes a particular value equals 0 can only calculate the probability that a random variable falls within a certain interval-the definition of a description distribution function that is applied to the probability distribution of two types of random variables by the x-axis and the area below the probability density curve: F (x) =p{x≤x} Probability distribution of common discrete random variables two-item distribution N-Bernoulli test: Only two possible outcomes for a single test are "success" represents the result of concern, and the opposite result is that the probability of "success" in each trial of "failure" is independent of the PN test. When the two-item distribution graph is p=0.5, when the two-item distribution is centered-symmetric p≠0.5, the two-item distribution is always asymmetric p<0.5 when the peak is at the left p>0.5 of the center, the peak is increased infinitely on the right side of the center with N. The significance of two distributions tending to the distribution of normal distribution Piosson the box contains 999 black pieces, a white pawn, in a sample, the probability of the white pieces of the smoke 1/1000 in 100 sampling, pumping in 1, 2, ... The probability of 10 white pieces is ... The number of radiation per unit of time of the radioactive substance count the amount of dust in the unit volume of the blood cells or microorganisms in the unit area under the microscope. Prevalence of non-communicable diseases with low prevalence in bacterial count population   characteristics: Distribution of rare event occurrences   "description" Historically, the Poisson distribution was introduced as a two-item distribution in 1837 by the French mathematician Poisson, and if the event with a small probability of success in the experiment is called a rare event, then the number of rare events in N heavy B-Test is approximate to the Poisson distribution when n is sufficiently large. At this point, the integer part of the parameter λ [happens to be the most likely number of rare events, and in practice the Poisson distribution is used as a mathematical model for the probability distribution of rare events in a large number of repetitive independent trials, such as unfortunate events, accidents, failures, non-common diseases, natural disasters, etc., are rare events.   Many random phenomena are subject to Poisson distribution. First, social life requirements for services: such as the number of calls to the telephone switch, the number of passengers arriving at the public station is approximately subordinate to the Poisson distribution. Another area is physics. The radioactive fission falls to the electrical point of an area;The radiation of the heat electron is subjected to poisson distribution.   The definition and graphic features of Poisson distribution   set random variable x all possible values are 0, 1, 2, ..., and the probability distribution is: 





It is often used to describe the random distribution law of the total number of rare "particles" in unit time, unit plane or unit space. The number of occurrences of rare events is x, then x is subject to Piosson distribution.   Two or two distribution and Poisson distribution   Historically, Poisson distribution was an approximation of two distributions, introduced in 1837 by French mathematician Poisson. In practice, many random phenomena obey or approximate the Poisson distribution. Example in He book, Feller discusses the statistics of flying bombhits in the south of London during the Second world War . Assume that's live in a district of size ten blocks by ten Blocksso that's total district are divided into small squa Res. howlikely is it, the square in which, you live would receive no hitsif the total area is hits by the bombs?  with X Indicates the number of bombs falling into the cell, then X~b (400,1/100)   n=400,p=1/100 so P (x=0) = (99/100) ^400 is approximated with Poisson distribution. X approximately obeys the Poisson distribution of the parameter 4 =np=400*1/100 i.e. x~p (4) so P (x=0) =exp ( -4) p (x=0) = (99/100) ^400 can be calculated (99/100) ^400= 0.01795055328exp (-4 ) = 0.01831563889 We refer to events in which the probability of a small occurrence in each trial is called a rare event. such as Luposon theorem, such as earthquakes, volcanic eruptions, catastrophic floods, accidents and so on, the number of rare events in N-heavy Bernoulli test is approximately subject to Poisson distribution .  three, General conditions of Poisson distribution   in nature and in people's real life, we often encounter some sort of event at random moments. The sequence of events that occur at random times is called a random event stream. If the event flow has smoothness, no effect, and general nature, The event stream is called a Poisson event stream (Poisson stream). The following is a brief explanation of stationarity, no effect, and general nature. Stationarity: In any time interval, the probability of K-Times (k≥0) of an event is only dependent on the interval length and the interval endpoint is notClosed. No effect: the occurrence of events is independent of each other during periods of non-overlap. General: If the time interval is sufficiently small, the probability of two or more occurrences of the event is negligible. For example, the number of а particles emitted by a radioactive source; Number of calls received by a telephone exchange; number of aircraft landing at an airport; The number of customers a shop assistant receives; the number of broken ends of a spinning machine can be regarded as a Poisson flow. For Poisson flow, the number of events (such as traffic accidents) that occur in any time interval (0,t) is subject to the Poisson distribution with the Λt parameter. λ is called the strength of the Poisson stream. The central limit theorem of independent distribution (also called Levi I Lindberg theorem) is set X1, X2, ... is a sequence of independent and distributed random variables with finite μ and variance σ2 (i=1,2,... When n→∞, the above theorem shows that the distribution of the N-sum of the random variable sequences with independent distributions tends to be normally distributed, no matter what distribution they obey. It can be concluded that no matter what distribution the population obeys, as long as its mathematical expectation and variance exist, when the sample quantity n is sufficiently large, it tends to normal distribution. This theorem lays a theoretical foundation for the sampling inference of mean value.   Example 11 stores using scientific management, by the store's past sales records know that the monthly sales of a certain commodity can be described by the Poisson distribution parameter λ=5, in order to ensure that more than 95% of the guarantee is not out of stock, ask the store at least a certain number of items at the end of the month? Solution: Set the number of sales per month for this product is X, known as the Poisson distribution of the parameter λ=5. Store at the end of the month should be entered into a certain commodity m pieces, so m+1=10,m=9 pieces N heavy Bernoulli the number of rare events in the experiment is approximately subject to the Poisson distribution. Poisson distribution in Management science, Operations research and some of the problems of natural science occupy an important position. Example 2: The probability of a child having a negative response to a hepatitis B vaccine is 0.001, and the probability of adverse reactions among 2000 children with 3 and more than two children is determined. p=0.323 Example 3: Insurance business is one of the earliest use of probability theory, insurance companies in order to estimate the profit, the need to calculate a variety of probabilities. Insurance companies now provide a life insurance for society, According to the information available, the risk of death associated with this insurance business in the population is 0.0020, and 2500 people are participating in this insurance, and each insured person will pay $120 per year on January 1, while the family member may receive $20000 in insurance from the company at the time of death. Ask: (1) What is the probability of the insurer losing money? (2) What is the probability that the insurance company will make a profit of not less than 100,000 yuan and 200,000 yuan? Solution: Each year January 1, the insurance company's income of 300,000 Yuan =120, if the death of a year, the insurance company to cope with 20000 yuan this year, so "company loss" means 20000 >300000 namely > 15, so the "company loses money" event is equivalent to "more than 15 people die in a year"of events, and thus the probability of "more than 15 people die in a year", if the "insured person in a year of death" as a random test, the problem can be n=2500,p=0.002 test to approximate, so p{insurance company profit not less than 100000 yuan}=0.986305p {The insurance company makes a profit of not less than 200000 yuan}=0.615961 Example 4: A district tax authority in order to make reasonable arrangements for the Tax Payment service window, it is necessary to investigate the customer arrival situation of a levy point in the area. The customer arrives at the Poisson distribution, and according to the observed results, there are 4 customers per 30 minutes on average. Q: (1) How much is the probability of arriving at 3 customers in 30 minutes? (2) What is the probability of reaching more than 4 customers within 30 minutes? Solution: The number of customers arriving within 30 minutes is the random variable x, which is set by the question X~p (4). (1) How much is the probability that 3 customers arrive in 30 minutes? P (x=3) =e^ (-4) *4^3/3!=0.1954. (2) What is the probability that 4 or more customers arrive within 30 minutes? P (x≥4) =1-p (x=0)-P (x=1)-P (x=2)-P (X-3) =1-0.0183-0.0733-0.1465-0.1954=0.5665 Example 5: In order to ensure the normal operation of equipment, need to be equipped with the right amount of maintenance personnel, a total of 300 units, Each work independently of each other, the probability of failure is 0.01, if in general, a device failure can be handled by one person, ask at least how many maintenance personnel should be equipped to ensure that when the equipment failure can not be timely maintenance of the probability of a small 0.01? If one person is responsible for 20 units, what is the probability that the 20 units will fail and not be processed in time? If 3 people are responsible for 80 units, the probability that the 80 equipment failure can not be dealt with in a timely manner (the average person is responsible for 27 units) Solution: Set X for 300 devices at the same time failure of the number of units, x~b (n,p), n=300, p=0.01 to be equipped with n maintenance personnel, the request is to meet P (x>n) < 0.01 minimum N.P (x>n) = 1-p (x≤n) <0.01p (x≤n) ≥0.99λ=np=300*0.01=3 only 8 workers at this time, X~b (20,0.01) The probability is P (x≥2) =1-p (x< 2) =1-p (x=0)-P (x=1) ≈0.017523λ=np=20*0.01=0.2 If 3 people are responsible for 80 units, the probability that 80 devices fail and cannot be processed in time (the average person is responsible for 27 units) X~b (80,0.01) λ=np=80*0.01= 0.8P (x≥4) ≈0.0081

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