We see that probability theory is really just a reduction of common sense to calculation, which allows us to evaluate with a rational mind precisely certain intuitionistic insights that often fail to explain. It is remarkable that probability theory, a science that has its roots in thinking about games of chance, What are the most important questions in life are largely a matter of probability theory. — Pierre-Simon Laplace, French mathematician and astronomer

“Theory of probability and mathematical statistics” is one of the common learning in science and engineering courses, but after the students in learning, seems to have only learned some mathematical formulas, did not produce an intuitive understanding of statistical concepts as and the understanding of the theory of the method according to the method (which is related to the application conditions and limitations), subjective probability and objective probability points not clear. Mathematical statistics and probability theory are two sister disciplines that are closely related. In general, probability theory is the basis of combing statistics, while mathematical statistics is an important application of probability theory.

Mathematical statistics is a subject with strong application, which has its method, application and theoretical basis. In the West, the word “mathematical statistics” refers specifically to the mathematical basis theory of statistical methods, but in China, it has a wider meaning, including methods, applications and theoretical basis, which is called “statistics” in the West. In Our country, it is sometimes necessary to use the two terms differently, because there is still a branch of statistics that is recognized as a social science. — Probability Theory and Combing Statistics

preface

My question is why 0.75 is just right. Then I read on Zhihu that in fact, high-level languages also have data structures similar to HashMap, such as 0.65 in Go and 0.8 in Dart. 0.762 in Python.

So why is HashMap 0.75 in Java? I found the answer in a comment in the HashMap:

Because TreeNodes are about twice the size of regular nodes, we use them only when bins contain enough nodes to warrant use (see TREEIFY_THRESHOLD). And when they become too small (due to removal or resizing) they are converted back to plain bins. In usages with well-distributed user hashCodes, tree bins are rarely used. Ideally, under random hashCodes, The frequency of nodes in bins follows a Poisson distribution (en.wikipedia.org/wiki/Poisso…). With a parameter of about 0.5 on average for the default resizing threshold of 0.75, although with a large variance because of resizing granularity. Ignoring variance, The expected occurrences of list size k are (exp(-0.5) * POw (0.5, k)/factorial(k)).

Since tree nodes are usually twice as many as regular nodes, we use trees when the number of buckets (entries in buckets) exceeds TREEIFY_THRESHOLD, and when the number of buckets becomes small due to expansion or removal, the tree structure is rarely used if the hash distribution is uniform. Ideally, in random testing, the default expansion factor is 0.75 and the distribution of nodes in the bucket (frequency of occurrence) follows the parameter 0.5. Although the granularity of the expansion will be quite different. Ignoring the variance, the probability of placing K elements in a given index in the array is: P(X=k)=0.5kk! E – 0.5, k = 0, 1,… P (X) = k = \ frac {^ 0.5 k} {k! } e ^} {- 0.5, k = 0, 1,… P(X=k)=k! Ke 0.5-0.5, k = 0, 1,…

But found that I am still in the concept of cognitive error, similar to when I was walking to see a 0, approached a look is 8, just stay away from the never see clear, pay attention to the above 0.75 is preset conditions in front, the back of the poisson distribution is used to explain why when the bucket of the elements in the more than eight chain table to convert a red-black tree. Fortunately, there is a formula found on StackOverFlow that can be deduced:

  • What is the significance of load factor in HashMap?

But there’s also probability involved.

For example, when you calculate average time complexity, if you’re looking for a single linked list, you multiply the probability of finding times the number of searches, which is what you expect in probability theory.

For example, when I wanted to check the quality of a batch of cards, I adopted a plan to fold each card hundreds of thousands of times to prove that the quality of this batch of cards is really qualified, but some students would naturally say that this kind of test is not reliable, the reason is that if this is broken at 6001 times, then this batch of cards will not be invalid? The presupposition of this rebuttal is that my card is memorized, that is, the last quality verification will remain, but my card is not memorized, and I have made it clear that this is the case. This thinking comes from a video I saw when SCROLLING Douyin, which said how many times do we fold each card in half before leaving the factory to ensure good quality. The bottom line is that the person who gets the card also folds it in half slightly once, so it is not the card if it is broken. And one of the comments at the bottom that really struck me was, this card has no memory, it’s a Poisson distribution.

For example, after learning the relationship between normal distribution and the central limit theorem in college, I still don’t understand the connection. In short, there are a lot of data in the real world that obey normal distribution, so we naturally ask why? The central limit theorem says, roughly speaking, that a large number of random variables are approximately normally distributed.

Such as vaccine effectiveness is how to come out, I want to be a period of video to explain, because the last outbreak and the discussion of the classmate, my classmates for the vaccine is full of distrust, at that time, my classmate’s argument is that SARS to now also does not have developed vaccines, analogy analysis, the new crown so fierce, new crown developed so quickly and less reassuring. However, in my opinion, analogical analysis is not always correct. For example, without the time dimension, a house in the same position ten years ago may be available ten years ago, but it may not be available ten years later. There are factors we can’t see this in the inside, the inertia of the human based on their on their own experience to judge another deep understanding of the their fields, for this I want to do video of popular science and popular science article, what do you want to be the idea of video of popular science and popular science articles from roughly zhihu began a question and answer: the teacher taught you what are the most wonderful work knowledge?

Several replied that the human mouth has more bacteria than the toilet, so the mouth is dirtier than the toilet. Couples kiss… These are all very good. My comment is:

I don’t feel like I’m a medical student, like bacteria equals dirt. In the toilet, most of the bacteria belong to the actinobacteria, bacteroidetes and other categories, these bacteria are very harmful to the human body, and some can even enter the air through the air flow of the toilet. Although the human body saliva bacteria, but most are harmless. What I want to say is that being dirty is not only a matter of the quantity of bacteria, but also a matter of the quality of bacteria. What’s more, I just said that the teacher said that people should be subject to the light of their own inner rationality, rather than as if what the teacher said was all right. We all know this truth, but we all believe in authority. I am disgusted with the idea that dirty is defined by the number of bacteria. Salmonella has two deadly forms, including salmonella enterica and Salmonella typhimurium. Salmonella typhi causes typhoid, a disease that kills 216,000 people a year. It is spread through an infection of the face and urine. There are about 500 to 1,000 different species of bacteria in each human body, and they can produce about 100 trillion individual cells in an adult — about 10 times as many as a person’s entire body cells. Even if you shower every day, these germs won’t get rid of. According to you, a clean person is dirtier than a petri dish containing salmonella. (If you don’t believe me, we all have more bacteria on our bodies than a toilet does, so we’re all dirty. Is the kind of wash can not wash off the dirty, is the kind of even become a corpse is more dirty than the toilet of the dirty) dirty is a perceptual understanding, your teacher is at least a graduate degree, this do not understand? I always thought it was rumor, not knowledge. And do you have an engineering background? Seriously doubtful! I’m sorry if I offended you

For example, in HashMap, the length of the hash bucket array table must be 2 to the power of n (must be composite), which is an unconventional design. The conventional design is to design the size of the bucket as a prime number. Relatively speaking, prime numbers are less likely to cause conflicts than composite numbers, and you see the probability again. So what is probability?

What is probability?

Subjective probability

The word probability appears frequently in the real world. People often use probability to express the measurement of whether a certain event has happened. For example, in “Fighting through the Sky”, Xiao Yan swallowed the heart fire of the green lotus, which was described as follows:

According to rough calculation, without these auxiliary items, the success rate of phagocytic fire is less than 1%, but with them, the success rate may be raised to about 10%.

Such that we can find a lot of, in the real world when the weather is cloudy, for example, people will generally said, I have ninety percent sure it’s going to rain today, ninety percent of possibility is Shakespeare wrote Hamlet really, Oswald the possibility of the assassination of President Kennedy alone is eighty percent.

But is this probability in mathematics? It’s not, to put it simply, a measure of how confident people are about what they say. In other words, less than one percent is that Xiao Yan is not so confident about whether he can successfully devour the fire. Ninety percent is sure that it will rain today means that we are very sure that it will rain today. Probability as individuals that degree of measurement is often referred to as subjective probability, not like a coin toss experimental frequency as the probability of support, is a kind of how confident the measures, the number of different people give difference not much meaning, like overcast today, do you think there will be a eighty-five percent chance it will rain, and I think it’s about ninety percent. The only explanation for this is that I’m more sure it’s going to rain today.

And objective probability, probability in mathematics is with frequency to hang together, we always give first sample space, the assumption of all possible outcomes is known, and we don’t know the result before experiment, a collection of all possible outcomes constitute what we call the sample space, and any subset of the sample space we call event, An event is a collection of some result of an experiment. If the result of the experiment is contained in E, then event E is said to have occurred.

Compared with the well-known experiment of coin toss and dice roll, we can easily get the sample space. For coin toss, the sample space is as follows:

  • Face up (digital face)
  • Back up (non-digital face)

So if I were to say what are the probabilities of heads? May have a society without answer half, or fifty percent, but fifty percent of the hypothesis that the answer lies in the COINS are uniform, that both the front and the possibility of equal, people tend to be in the end after the experiment of the sample space, take the events in the sample space, divided by the total number of the sample space, the number of in order to calculate the probability of a certain event occurs. Writing here, I recall a discussion I had with one of my college classmates at the beginning of the course Probability theory and Mathematical Statistics. My classmate came to the following conclusion:

  • I have a 50/50 chance of failing this course (because there are only two events in the sample space: fail, fail)
  • There is also a 50/50 chance of being a supervisor (because there are only two in the sample space: yes and no)

Conclude that the cause of the large is why do much to solve the problem, many probability given the title of the default probability, such as the subject, math subject and the real world, will be the default and default possibilities, such as the above error judgment is that the start will “fail”, “monitor” when the coin is a homogeneous structure, Predicting a 50/50 split between failing and failing, being class president and not being class president, it follows that any single event in the sample space with only two situations will have a 50/50 chance of occurring. Have such a coincidence in the theory of problem sets, the number of subsets in sample space divided by the total number of the sample space is just equal to the probability of the event (such as possible sample space), such as dice, a total of six kinds of circumstances, we generally assumptions are dice uniform texture, the implied conditions comes on each face are such possibility, And then we deduce that each face has a one in six probability, which is what we call in probability theory the equal probability sample space.

The wrong conclusion obtained above lies in the assumption of equal possibility at the beginning. Taking the second one as an example, we have no way to know the possibility of events occurring in the sample space of class monitor. My personal inference is that the probability of being elected is 0.0000001%. Strictly speaking, I have also given the possibility of this model again, but HERE I just think it is impossible for my college classmate to be elected class president.

Subjective probability can be understood as a state of mind or inclination, which has two roots. First, it is based on experience and knowledge. For example, when I was young, my family was drying wheat on the highway, and that evening it was cloudy, SO I concluded that it would rain today. But a grandfather in the same village told my father, don’t take it, the weather is good today, the wind is good, it won’t rain. My father finally listened to the grandfather, and although the sky was overcast, there was no rain. From this point of view subjective probability also has an objective background, after all, different from idle gossip. The second reason is that taking the weather in Shanghai today as an example, it was raining yesterday, but WHEN I went out for breakfast, I still didn’t take my umbrella with me. The reason is that EVEN if it was raining, I didn’t care, and taking an umbrella makes me feel a bit troublesome.

Subjective probability is characterized by the fact that it is not accepted on a solid objective basis and therefore seems to be repudiated by science (which seeks truth). It is not clear how to understand subjective probability, but do not deny it comprehensively. One reason is that the concept has a broad basis in life. We estimate the probabilities of situations almost all the time, and different people rarely agree on an “objective” basis. The other reason is that it may also reflect a tendency of the cognitive subject, which has its social significance. For example, asking “Will the economy be better off three years from now? People of different economic status, social status and even political orientation make different estimates. There may not be much truth in individual estimates; But on the whole, it reflects the broad masses of society’s confidence in long-term development. This is useful information for sociologists and even policy makers. Third reason is that in the design () economic and other interests gain and loss of decision making problems, in how many different people in a different position and grasp the information, the size of the possibility of certain events to refer to these circumstances and consequences to measure, suitable for someone’s decisions, such as less risky decisions, may also not suitable for another person, for another person, This decision may be risky. So the concept of subjective probability also has a practical basis. In fact, many decisions involve personal judgment, and this is subjective probability

Objective probability

Experiments and events

The subjective probability we mentioned earlier is a state of mind or a predisposition as to whether or not an event will happen, so let’s look at the term “event”, what is an event? In the usual sense, it usually refers to something that has happened, such as the xyz plane crash, the Japanese attack on Pearl Harbor in 1941 and so on. In probability theory, on the other hand, an event is not something that has happened, but a “statement” of something (or something) that may or may not have happened and will not be known until the results of the relevant “experiment” are known.

If it’s not going to rain before 6 o ‘clock today, we’re not saying it’s already happened, of course, but whether it’s going to happen or not depends on the results of the experiment, which is looking at the weather before 6 o ‘clock in the afternoon.

By extension, we would not be difficult to understand: in probability theory, the general meaning of the word “event” is that: (1) there is a clearly defined, “experiment”, the word, has active meaning, and as the above example, people just out of passive position, just the record without intervention meteorological process, this kind of circumstance commonly referred to as the “observation”. In statistics, which sometimes has practical meaning but is not important for the present discussion, the word “experiment” can be included as observation. (2) The full range of possible outcomes of the experiment were specified before the experiment. For example, all the possible outcomes of the experiment are only two: first, it won’t rain before 6 o ‘clock today, which is represented by A, and second, it will rain by 6 o ‘clock today, which is represented by A (4) ‾\overline{A}A}. Therefore, this experiment can be written as {AAA, A (4) overline{A}A}. 4, (the experiment is not finished, that is, until 6 p.m.). If not A{A}A, it is an ‾\overline{A}A.

But in many cases, we do not know exactly all the possible outcomes of an experiment, but we can know that it does not exceed a certain range. This range can also be used as the full range of possible results of the experiment. As in precedent, if we are interested in more than is that if it will rain before 6 PM, but need to record rainfall before 6 PM (in mm) for the unit, is the result of the experiment will be negative real number x, we are unable to determine the possible value range of x, but you can take for this range of [0, up \ infty up], it can always contains all the experimental results, Although we know that certain outcomes, such as x > 10000, are unlikely to occur. We can even call this range (- ∞\infty∞, ∞\infty∞), and there’s a certain mathematical abstraction here, which can be very handy, as we’ll see later.

(3) We have an unequivocal statement that defines A certain portion of the total number of possible outcomes of the experiment, or rather, A certain portion is called an event, as in the following example, A is the certain portion of all possible outcomes (AAA, A \overline{A}A). In the dice roll example, we can define many events, such as:

  • E1 = {roll even points} = {2,4,6}
  • E2 = {throw prime points} = {2,3,5}
  • E3 = {roll multiple of 3} = {3,6}

And so on, they clearly define the set of total experimental results (1,2… A corresponding part in,6.

If we do the experiment again, now that the dice once, when throwing the results for 2 or 4, or 6, we say event E1 “happened”, otherwise E1 “doesn’t happen,” said event as a result, we also can say: the event is related to the experimental results of proposition, its correct or not depends on how the result of the experiment.

In probability theory, a single experimental result is sometimes called a “fundamental event.” In this way, one or a number of basic events together constitute an event, and the basic event is itself an event. In the dice case, there are 1,2… ,6, etc. 6 basic events. Event E2 is composed of 3 basic events 2,3 and 5.

Imagine you’re in a situation where you’re rolling a die, and if there’s a prime point, you’re going to win, and before you roll the die, you’re going to think: I’m going to win or not, it depends on chance. Therefore, in probability theory, events are often referred to as “random events” or “chance events.” Random simply means that whether or not an event occurs in a given experiment depends on chance, with extremes being “certainties “(things that are certain to happen in the experiment, such as rolling a die with a roll of 6 or less) and” impossibilities “(times that are unlikely to happen in the experiment). There is no chance in either case, but for the sake of convenience, consider them exceptions to a random event.

Probability definition of classical probability

Continuing with the previous paragraph, suppose that an experiment has a finite number of possible outcomes E1, E2… Given the conditions of the experiment and the methods of its implementation we find no reason to think that one of the results such as EI is more likely to occur than any of the other results such as EK we have to assume that all results e1… The result of an experiment in which en has an equally likely chance of appearing, i.e., a 1/N chance of appearing, is often called “equally likely”.

Take the example of rolling dice, if: (1) the dice material is absolutely uniform; ② Dice are absolute regular hexahedron; ③ If the dice roll is sufficiently high from the ground, it is generally agreed that the chances of each side appearing should be equal. In real life, of course, this is only an approximation, and the number of points engraved on the die also affects its symmetry.

On the basis of the concept of “equal possibility”, the definition of classical probability is naturally introduced: Suppose an experiment has N equally possible results, and event E contains exactly M of them, then the probability of event E is denoted as P(E) and defined as:

P(E) = M / N

The limitations of classical probability are obvious: it can only be used when there are a finite number of equally likely outcomes, but in some cases the concept can be extended slightly to include an infinite number of outcomes, known as “geometric probability.”

Definition of probability from statistics

From a practical point of view, the statistical definition of probability is nothing more than a way to estimate the probability of events by experiment. Take the example of “rolling dice”. If the dice are not uniform cubes, the probability of each side appearing in the roll need not be the same. What is the probability of E1 at the time of “6” can no longer be determined by a theoretical consideration alone. But we can do an experiment: roll this die over and over again a large number of times, say n times, and if there is a 6 in n rolls and a total of M1, then m1 over n is E1 in n experiments (each roll counts as one experiment). The gist of the statistical definition of frequency is to say, let’s take this frequency m1/n as an estimate of the probability P of E1. The intuitive background for this probability is simple: the probability of an event’s occurrence should be characterized by its frequency over repeated experiments.

The general case is no different from this, except that in the preceding statement, “roll the dice” is replaced by some general experiment, and the “occurrence of 6” event E1 is replaced by some general experiment. The point is that the experiment must be repeated in large numbers under the same conditions so that it is possible to observe the frequency of the event.

The shortcoming of the above definition, that is, frequency is only an estimate of probability rather than probability itself, can be solved by the following statement formally:

The probability of event E is defined as a number P with the property that the frequency of E oscillates near P as the experiment is repeated and the oscillation becomes smaller and smaller as the number of repetitions increases, or simply that the probability is the limit of the frequency as the number of experiments increases indefinitely. To do so, the question must be answered: How can you prove the existence of P with the above properties, or that the existence of P is a hypothesis? Proponents of frequency definitions often answer this question by saying that n(E)/n tends to some constant limit as an assumption, or an axiom, of the system as a whole. But this assumption seems extraordinarily complicated because, despite the fact that it is necessary to assume that a limit of frequency exists, it is not the most basic or simple assumption. At the same time, this assumption is not shared by everyone. Wouldn’t it make more sense to define some simpler, more obvious axioms, and then prove that the frequency in some sense approaches the limit of a constant? This is also the axiomatic method of modern probability theory (implemented by the great Soviet mathematician Kolmogorov)

The significance of the statistical definition of probability is not that it provides a way of defining probability, which it does not, because you can never define exactly what the probability of any event is based on this definition. Its importance lies in two aspects: first, it provides a method to estimate probability. This has been mentioned above. There are many applications. For example, in a population sampling survey, the illiteracy rate of the entire population is estimated based on a small sample of people; In industrial production, the rejection rate of products is estimated according to some products extracted. Second, it provides A criterion to test whether the theory is correct or not. Suppose that the probability P of an event A is calculated according to certain theories and assumptions, etc. We are not sure whether the theory or assumption is consistent with the reality, so we resort to experiments, that is, we conduct A large number of repeated experiments to observe the frequency M/N of event A. If m/ N is close to P, the experimental results are considered to support relevant theories. If you want to go farther, you think the theory may be wrong. This kind of problem belongs to an important branch of mathematical statistics – hypothesis testing.

Mathematical expectation and mean time complexity

In the introduction of this name, let’s see a game, a, B two people gambling skills are the same, bet 500 yuan everywhere, agreed to win three rounds for the first win, get all 1000 yuan. Now a win two games, b win a game under the circumstances of suspension, ask how gambling should be divided? Let’s analyze the bet. It’s unfair for a to divide equally, and unfair for b to divide all. One seemingly acceptable method is to score by the number of wins, with player a taking 2/3 and player B taking 1/3. Careful analysis, this division is still very reasonable, imagine if read two rounds, the result is nothing more than the following two situations:

  • A nail
  • Party a and party b
  • B a
  • B b

“A b” means a win in the first round and b win in the second sentence. And so on. When we combine the three games we have already bet with the situation above (i.e., player A and Player B finish five games), we see that player A wins the first three games, because player A gets $1000, and only player B gets $1000 for the last game. The four scenarios listed above should be equally likely, given the same betting skills. Therefore, the probability ratio of party A and Party B to win is 3: 1. All bets should be divided according to this ratio, namely, Party A gets 750 yuan and Party B gets 250 yuan. The formula is as follows:

1000×341000\times \frac{3}{4}1000×43 + 0×140\times \frac{1}{4}0×41 = 750 1000×141000\times \frac{1}{4}1000×41 + 0 x 340 \ times \ frac {3} {4} 0 x 43 = 250

If a random variable X is introduced, X is equal to the outcome of the above scenario (two wins and one loss for A player), then X has two possible values: 1000 and 0, with probabilities of 34\frac{3}{4}43 and 14\frac{1}{4}41 respectively. And the expected value of a, the expected value of X, is going to be equal to

The sum of the possible values of X with their probabilities.

That’s where we get the name mathematical expectation, which comes from gambling, and it doesn’t sound very common or easy, and it’s not a very good name, but it’s been around in probability theory for a long time, and it’s been accepted, and it’s taken hold. The other name for it is mean, which is very easy to understand and very common. This will be explained below. Is that still a little abstract, and we need to be abstract and intuitive, and look closely at this formula, and see if there’s a sense of weighted average. Weight is the probability, and weight is the possible value of X.

Another interpretation of expectation comes from the frequency interpretation of probability. This interpretation (we’ll see the strong number theorem later) says that if an infinite number of independent repetitions are performed, then for any event E, the ratio of times E occurs is P(E). Suppose the possible values of random variable X are x1, x2… , xn. And the corresponding probabilities are P(x1),P(X2)… ,P(Xn), and we interpret the random variable as winning units in a game of chance. That is, each game we use xi units with probability p(xi), I = 1,2…… , n. Now let’s use frequency to explain that if we play this game continuously, then the ratio of our wins to xi is p(xi). Since this is true for all I (I =1,2… ,n), so our average game unit per game is:

Nxip ∑ I = 1 (xi) \ sum_ {I = 1} ^ {n} x_ip nxip (∑ x_i) I = 1 (xi) = E (X) E is the abbreviation of Expectation

Roughly speaking, we can think of the expectation, which we’ll call the mean, as the average win, and if the average win is less than your bet, there’s a good chance that you’re losing money, because the game itself is probably unfair. Then we go back to the average time complexity of the algorithm, and we simply introduce a problem:

For an ordered sequence, what is the average time complexity of sequential searches? This is actually an application of mathematical expectations, namely, finding the mean. Let’s give a simple example to discuss: {1,2,3,4,5}. If you look for 1, you’ll find it immediately, if you look for 2, you have to look twice, in the second position. When we search, we can’t determine which number to look for, and this involves the mean, assuming that the probability of finding any number in an ordered sequence is equal, Mathematical expectation of ∑ I = 1 n1n * (n + 1) * (n = (n + 1) 2 \ sum_ {I = 1} ^ {n} \ frac {1} {n} \ times \ frac {(n + 1) \ times n} {2} = \ frac {(n + 1) } {2} nn1 ∑ I = 1 x 2 (n + 1) (n = 2 (n + 1) (n + 1) * n2 \ frac {(n + 1) \ times n} {2} 2 * n (n + 1) to find out the number of the number when you compare the number. The mathematical expectation at this point is the average number of comparisons. This is the average complexity of the algorithm. Roughly speaking, the time complexity is how the operation time increases as n increases. We can say that the sequential search complexity increases linearly because (n+1)2\frac{(n+1)}{2} 2(n+1) is a linear function.

To summarize

This is the first chapter of probability theory and mathematical statistics, vaccine science popularization, data structure and algorithm analysis. I did not introduce a lot of mathematical formulas, but tried to introduce probability and sort out statistics in an intuitive way. We should not only be able to abstract, but also restore the abstraction to intuition. Roughly speaking, probability is an analysis of, or prediction of, an outcome given a known cause. Mathematical statistics, on the other hand, analyze causes when the results are known. This post was supposed to be called study notes, but writing it doesn’t feel like study notes. I am a mixture of “Basic course of Probability Theory” and “Probability theory and Mathematical Statistics”, because the later will be sent “Probability theory and mathematical statistics” learning article, so this is a continuation of the past, let’s call it an introduction.

The resources

  • Introduction to Probability Theory by Sheldon M. Rose, translated by Tong Xing, Wei Liang, Baosheng
  • Chen Xiru, probability theory and Mathematical statistics
  • Zhou Xingni, Ren Zhiyuan, Ma Yanzhuo, Fan Kai. New Perspectives on Data Structure and Algorithm Analysis
  • Why is the default load factor for hashMap 0.75? It should be a compromise between space and time. What is the statistical principle behind it?
  • What is the significance of load factor in HashMap?
  • Can’t understand Poisson part of Hash tables from Sun documentation
  • HashMap defaultLoadFactor = 0.75 has no relation to the Poisson distribution
  • Why is the loadFactor of HashMap 0.75