Forms
Two different versions of the Law of Large Numbers are described below; they are called the Strong Law of Large Numbers, and the Weak Law of Large Numbers. Both versions of the law state that – with virtual certainty – the sample average
converges to the expected value
where X1, X2, ... is an infinite sequence of i.i.d. integrable random variables with expected value E(X1) = E(X2) = ...= µ. Integrability means that E(|Xj|) < ∞ for j=1,2,....
An assumption of finite variance Var(X1) = Var(X2) = ... = σ2 < ∞ is not necessary. Large or infinite variance will make the convergence slower, but the LLN holds anyway. This assumption is often used because it makes the proofs easier and shorter.
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
Read more about this topic: Law Of Large Numbers
Famous quotes containing the word forms:
“It is not however, adulthood itself, but parenthood that forms the glass shroud of memory. For there is an interesting quirk in the memory of women. At 30, women see their adolescence quite clearly. At 30 a woman’s adolescence remains a facet fitting into her current self.... At 40, however, memories of adolescence are blurred. Women of this age look much more to their earlier childhood for memories of themselves and of their mothers. This links up to her typical parenting phase.”
—Terri Apter (20th century)
“Being the dependents of the general government, and looking to its treasury as the source of all their emoluments, the state officers, under whatever names they might pass and by whatever forms their duties might be prescribed, would in effect be the mere stipendiaries and instruments of the central power.”
—Andrew Jackson (1767–1845)
“I may not tell
of the forms that pass and pass,
of that constant old, old face
that leaps from each wave
to wait underneath the boat
in the hope that at last she’s lost.”
—Hilda Doolittle (1886–1961)