Probability Theory - Law of Large Numbers

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

\begin{array}{lll} \text{Weak law:} & \overline{X}_n \, \xrightarrow{P} \, \mu & \text{for } n \to \infty \\ \text{Strong law:} & \overline{X}_n \, \xrightarrow{\mathrm{a.\,s.}} \, \mu & \text{for } n \to \infty . \end{array}

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:

    “There is ... but one response possible from us: Force, Force to the uttermost, Force without stint or limit, the righteous and triumphant Force which shall make Right the law of the world and cast every selfish dominion down in the dust.”
    —Woodrow Wilson (1856–1924)

    “Half the testimony in the Bobbitt case sounded like Sally Jesse Raphael. Juries watch programs like this and are ready to listen.”
    —William Geimer, U.S. law educator. New York Times, p. B18 (January 28, 1994)

    “Whenever you pray, make sure you do it at school assemblies and football games, like the demonstrative creatures who pray before large television audiences. That is the real goal of the thing. But do not, I urge you, pray all alone in your home where no one can see. That does not get you ratings.”
    —Garry Wills (b. 1934)

    “Publishers are notoriously slothful about numbers, unless they’re attached to dollar signs—unlike journalists, quarterbacks, and felony criminal defendents who tend to be keenly aware of numbers at all times.”
    —Hunter S. Thompson (b. 1939)