About the law of tremendous quantities (zbch) is scribbled apparently invisibly (for example, in English, here and here, yes). It is here that in the text I will try to tell you about what the law of tremendous quantities is not classified - about the misperception of this law and the potential pitfalls hidden in mathematical formulations.
Let's activate with what is the canon of huge numbers. Informally, there is such an exact axiom about the fact that "the probability of deviations of the median in the sample from the mathematical expectation is small" and, most characteristic, "this possibility tends to zero near the increase in the sample." Absolutely informal, the axiom states that we are given the opportunity to be in the necessary stage, we are sure that the mean after our sample is rather close to the "real" mediocre and therefore describes it well. Of course, this implies the existence of a classical statistical "baggage" - our observations from the selection are appropriate to characterize the same phenomenon, they must exist independently, and the idea that there is some kind of "real" arrangement with a "real" mean is not can ignite our substantial doubts.
When formulating the law, we tell the "average for the sample", that's all that can exist is precisely fixed as if such an average, occurs near the effect of the law. For example, a portion of incidents in a corporate heap may become recorded as an average - we just need to write down the existence of an action as "1" and the absence as "0". In general, the average will be the same frequency and the frequency should be close to the abstract average. This is why we expect that the share of "heads" in the toss of a perfect coin will be close to ½.
Let's examine today the traps and false images about this law.
First, the ZBCH is sometimes correct. This is just an extremely accurate axiom with "input data" - assumptions. If the assumptions are wrong, then the law is not obliged to be fulfilled. For example, this is the case if the observations are dependent, or if there is no certainty that the "real" mean is available, of course, or if the event under investigation is modified in time and we cannot say that we notice the same value. To be honest, in the set stage, the ZBCH is also correct in such cases, for example, for the sake of weakly correlated observations or, in that case, sometimes the contemplated measure is modified in time. However, for the sake of respectfully adding the given to concrete reality, a well-trained specialist mathematician is needed.
Second, it appears to be true that the ZBR states "the post-sample mean is close to the true mean." However, such a statement remains incomplete: one must unconditionally add “with a noble part of probability; and this opportunity is constantly less than 100%. "
Third, one wants to construct the ZBP as "the sample mean collides with the full-fledged mediocre with unlimited sample growth." However, this is not true, therefore, that the sample average does not converge anywhere, the causality is random and remains so for the sake of a different sample size. For example, if you deliver an invariant coin a million times, everything is equal, there is a possibility that the share of eagles will be distant in ½ that is, equal to zero. In a sense, there is always the opportunity to inherit something unusual. We must admit, however, that our intuition still tells us that the ZBCH should characterize a certain similarity, and this is actually the case. It is not the mean that "converges" exclusively, but "the probability of the difference between the test median through its true value", and collides with zero. The causality of this idea is unconsciously infinitely favorable ("the chances of noticing something extraordinary tend to zero"), mathematicians have developed an independent character of convergence - "convergence after probability" to achieve the desired result.
Fourthly, the ZBCH does not remember well that sometimes the check average can be calculated rather close to the theoretical one. The canon of tremendous quantities exclusively postulates the presence of a conditioned phenomenon, it does not remember anything about that, sometimes it can be used. It turns out, for the paramount question from the point of view of practice - “can I recycle the ZBCH for my selection of size n? ", The canon of tremendous quantities does not answer. Conclusions for these questions are provided by other theorems, for example, the Main Bounding Theorem. It gives an idea of where the test mean may deviate from its present value.
Finally, it should be noted the main significance of the ZBP in statistics and the concept of probabilities. The chronicle of this law began then, sometimes experts noticed that the frequencies of some repetitive phenomena stabilize and cease to change dramatically, if the condition of repeated resumption of the experiment, that is, observation, is observed. It was surprising that this "stabilization of frequencies" was tracked for completely unrelated actions - playing through the mooring
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