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:

    “The law of nature is, do the thing, and you shall have the power: but they who do not the thing have not the power.”
    —Ralph Waldo Emerson (1803–1882)

    “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)

    “This was the Eastham famous of late years for its camp- meetings, held in a grove near by, to which thousands flock from all parts of the Bay. We conjectured that the reason for the perhaps unusual, if not unhealthful development of the religious sentiment here, was the fact that a large portion of the population are women whose husbands and sons are either abroad on the sea, or else drowned, and there is nobody but they and the ministers left behind.”
    —Henry David Thoreau (1817–1862)

    “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)