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 large numbers, law of, law, large and/or numbers:

    “Always in England if you had the type of brain that was capable of understanding T.S. Eliot’s poetry or Kant’s logic, you could be sure of finding large numbers of people who would hate you violently.”
    —D.J. Taylor (b. 1960)

    “But what is classification but the perceiving that these objects are not chaotic, and are not foreign, but have a law which is also the law of the human mind?”
    —Ralph Waldo Emerson (1803–1882)

    “The law of God is a law of change, and ... when the Churches set themselves against change as such, they are setting themselves against the law of God.”
    —George Bernard Shaw (1856–1950)

    “Switzerland is a small, steep country, much more up and down than sideways, and is all stuck over with large brown hotels built on the cuckoo clock style of architecture.”
    —Ernest Hemingway (1899–1961)

    “Out of the darkness where Philomela sat,
    Her fairy numbers issued. What then ailed me?
    My ears are called capacious but they failed me,
    Her classics registered a little flat!
    I rose, and venomously spat.”
    —John Crowe Ransom (1888–1974)