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:
“From the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.”
—Charles Darwin (1809–1882)
“And what other great nation has statutes and ordinances as just as this entire law that I am setting before you today? But take care and watch yourselves closely, so as neither to forget the things that your eyes have seen nor to let them slip from your mind all the days of your life; make them known to your children and your children’s children...”
—Bible: Hebrew, Deuteronomy 4:8,9.
“If the hero is not a person, the emblem
Of him, even if Xenophon, seems
To stand taller than a person stands, has
A wider brow, large and less human
Eyes and bruted ears: the man-like body
Of a primitive.”
—Wallace Stevens (1879–1955)
“The forward Youth that would appear
Must now forsake his Muses dear,
Nor in the Shadows sing
His Numbers languishing.”
—Andrew Marvell (1621–1678)