Law of Large Numbers
Common intuition suggests that if a fair coin is tossed many times, then roughly half of the time it will turn up heads, and the other half it will turn up tails. Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of heads to the number of tails will approach unity. Modern probability provides a formal version of this intuitive idea, known as the law of large numbers. This law is remarkable because it is not assumed in the foundations of probability theory, but instead emerges out of these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence.
The law of large numbers (LLN) states that the sample average
of a sequence of independent and identically distributed random variables converges towards their common expectation, provided that the expectation of is finite.
It is in the different forms of convergence of random variables that separates the weak and the strong law of large numbers
It follows from the LLN that if an event of probability p is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards p.
For example, if are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p, then for all i, so that converges to p almost surely.
Read more about this topic: Probability Theory
Famous quotes containing the words law of, law, large and/or numbers:
“The law of nature is alternation for evermore. Each electrical state superinduces the opposite. The soul environs itself with friends, that it may enter into a grander self-acquaintance or solitude; and it goes alone for a season, that it may exalt its conversation or society.”
—Ralph Waldo Emerson (1803–1882)
“Who does not know history’s first law to be that an author must not dare to tell anything but the truth? And its second that he must make bold to tell the whole truth? That there must be no suggestion of partiality anywhere in his writings? Nor of malice?”
—Marcus Tullius Cicero (106–43 B.C.)
“No man, however benevolent, liberal, and wise, can use a large fortune so that it will do half as much good in the world as it would if it were divided into moderate sums and in the hands of workmen who had earned it by industry and frugality. The piling up of estates often does great and conspicuous good.... But no man does with accumulated wealth so much good as the same amount would do in many hands.”
—Rutherford Birchard Hayes (1822–1893)
“All ye poets of the age,
All ye witlings of the stage,
Learn your jingles to reform,
Crop your numbers to conform.
Let your little verses flow
Gently, sweetly, row by row;
Let the verse the subject fit,
Little subject, little wit.
Namby-Pamby is your guide,
Albion’s joy, Hibernia’s pride.”
—Henry Carey (1693?–1743)