Law of Large Numbers - University of Massachusetts...

Law of Large NumbersAlexandra Barbato and Craig O’Connell

Honors 391A Mathematical GemsJenia Tevelev

Jacob Bernoulli

Life of Jacob Bernoulli● Born into a family of important citizens in Basel,

Switzerland in 1623● Forced to study philosophy and theology by his parents● Graduated from the University of Basel in 1676● Studied mathematics and astronomy against his

parents’ wishes● Traveled around Europe meeting with many

mathematicians (including Boyle and Hooke)

Life of Jacob Bernoulli● Turned down an appointment in the

Church● Taught mechanics at the University of

Geneva from 1683 on● Studied Descartes, van Schooten, Wallis,

and Barrow● Began publishing in the journal Acta

Eruditorum which he established in Leipzig in 1682

Life of Jacob Bernoulli● Published a pamphlet on the parallels of logic and

algebra in 1685● Published work on probability in 1685● Published work on geometry in 1687

○ Construction to divide any triangle into four equal parts with two perpendicular lines

● Published Ars Conjectandi (The Art of Conjecturing) in Basel in 1713 after his death in 1705○ Law of Large Numbers

The Two Laws● Weak Law of Large Numbers (Bernoulli’s Fundamental

Theorem)● Strong Law of Large Numbers● If X1, X2, X3, … are independent random variables, and

each random variable has the same expected value μ, then the sequence of random variables given by the average of the random variables converges to μ as the sample size goes to infinity. (From: A Modern Theory on Random Variation)

Weak LawFor every ɛ > 0, as n goes to infinity, lim(PX)[(|((X1 + … + Xn)/n)-μ|) > ɛ)] = 0, where μ is the expected value of X

Strong LawTaking the limit as n goes to infinity, PX[lim(|(X1 + … + Xn)/n|) = 0)] = 1.

Bernoulli TrialA random experiment in which there are only two possible outcomes: pass or failure.Ex. Did the coin toss result in a heads?Ex. Was the child born a girl?

The Theory“When ratios of cases are not known a priori (prior to), it is possible to find them with as great a probability of correctness as one wishes a posteriori (afterwards), assuming that one is willing to collect sufficiently large amounts of data.”

The Situation“Imagine a situation in which some outcomes may happen in r cases and fail in s cases, such as a die rolling a 6. How many throws would be necessary to be confident with a high probability that the fraction of observed 6’s will be within a small interval of ⅙?”

The ProofBernoulli took high powers of the binomial (r + s)nt wherer + s = t. He expressed this as a series expansion. He then was able to prove that the interval around the largest term of the series corresponds to the probability that the observed ratio of outcomes will be less than (r + 1)/t and more than (r - 1)/t.

Simulated Dice RollsAverage Value on Dice:(1+2+3+4+5+6)/6 = 3.5

Simulated Coin Toss 1

Simulated Coin Toss 2Blue = HeadsRed = Tails

White and Black TokensBernoulli’s Example:● Three thousand white tokens and

two thousand black tokens in a pot● Take one out at a time, record the

color, and put the token back● If you do this enough times, you will

find the ratio of three white tokens for every two black tokens

Cracked Eggs● For every three-dozen eggs sold by a

grocer, an average of one of those eggs is cracked

● If we buy 12-dozen eggs, the likelihood that one for every three-dozen will be cracked increases

● If we buy 18-dozen eggs, the likelihood that one for every three-dozen will be cracked increases even more

CasinosWhile a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards an expected value over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game.

The Gambler’s FallacyPeople tend to think that if you have a string of one result, say a coin landing on heads 10 times in a row, that according to the LLN you must somehow get a sudden string of the opposite, flipping several tails in a row. This is not true however because the LLN is taken out of an infinite number of trials and this example is finite.

Diffusion 1

The concentration tends to balance out to an average as time goes on, however there are still random fluctuations.

Diffusion 2

Insurance● LLN is a basis of insurance● Insurance companies record and study the number of

claims over a large population● They can predict what proportion of 66 year old women

will die in a given year using LLN● They then plan rates accordingly

Works Cited1. http://www-history.mcs.st-and.ac.

uk/Biographies/Bernoulli_Jacob.html2. The Art of Conjecturing by Jacob Bernoulli3. A Modern Theory of Random Variation by Patrick

Muldowney4. Simulations from http://en.wikipedia.

org/wiki/Law_of_large_numbers