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:
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)
“Nor has science sufficient humanity, so long as the naturalist overlooks the wonderful congruity which subsists between man and the world; of which he is lord, not because he is the most subtile inhabitant, but because he is its head and heart, and finds something of himself in every great and small thing, in every mountain stratum, in every new law of color, fact of astronomy, or atmospheric influence which observation or analysis lay open.”
—Ralph Waldo Emerson (1803–1882)
“Nature has ordained that the man who is pleading his own cause before a large audience, will be more readily listened to than he who has no object in view other than the public benefit.”
—Titus Livius (Livy)
“The forward Youth that would appear
Must now forsake his Muses dear,
Nor in the Shadows sing
His Numbers languishing.”
—Andrew Marvell (1621–1678)