Poisson Process, Pokémons, and Prime Numbers

Poisson Process, Pokemons, and Prime Numbers

Kevin Wang

School of Mathematics and StatisticsThe University of Sydney

Last Modified: October 27, 2016

BackgroundThat French guy

Poisson ProcessDefinition

Coupon Collector’s Problem revisited

Number TheoryPrime Number Theorem and distribution of primes

Background

That French guy

The Poisson Distribution

Named after Simeon Denis Poisson.

P(X = k) =λke−λ

k!, k ∈ N. (1)

PP can be used to describe:

I radioactive decays,

I arrivals of buses

I or failures of Carslaw lift.(ongoing research)

Poisson Process

Definition

Counting process

Definition

A set of random variables {N(t)}t∈N is called a counting process if:

1. N(t) takes value in N.

2. N(0) = 0.

3. If s < t, then N(s) ≤ N(t).

4. For s < t, N(t)−N(s) equals the number of event in the interval (s, t].

Poisson Process

Definition

Poisson Process Definition

Definition

Furthermore, a Poisson Process has additional assumptions:

1. The random variable N(s+ t)−N(s) is Pois(λt) distributed, for all s, t ≥ 0.

2. Independent increments: the numbers of events that occur in disjoint timeintervals are independent.

I The time gap between any two consecutive arrivals (interarrival time) isExp(λ) distributed. P(X ≥ t) = 1− e−λt.

Poisson Process

Definition

Poisson Process

Definition

The Coupon Collector’s Problem with unequal probability

Problem

There are n individual coupons/Pokemons, each with capture probability pi.Assuming we can collect 1 coupon per unit time, what is the expected number ofcoupon do we need to complete the collection?

I We also derived that in the unequal probability case:

E(X) =

∫ ∞0

n∏i=1

(1− exp(−pix))

)dx. (2)

I The Poisson Process solution is much cleaner.

Poisson Process Solution to CCP (Sketch)

I Modelling the time of arrival of coupons instead of number of couponsneeded allow us to exploit independence of inter-arrival times.

I It can be shown that the expected time for collecting all coupons and thenumber of coupons needed in total are the same using properties ofinter-arrival times of PP.

Poisson Process Solution to CCP (Sketch)

I Modelling the time of arrival of coupons instead of number of couponsneeded allow us to exploit independence of inter-arrival times.

I It can be shown that the expected time for collecting all coupons and thenumber of coupons needed in total are the same using properties ofinter-arrival times of PP.

Poisson Process Solution to CCP

I This meant Z = max{Z1, . . . , Zn} is the maximum of a set of independentExp(pi) random variable:

P(Z ≤ t) = P(Z1 ≤ t, . . . , Zn ≤ t),

n∏i=1

P(Zi ≤ t),

n∏i=1

(1− exp(−pit)) . (3)

E(Z) =

∫ ∞0

P(Z > t)

∫ ∞0

n∏i=1

(1− exp(−pit))

)dt. (4)

Number Theory

BackgroundThat French guy

Poisson ProcessDefinition

Number TheoryPrime Number Theorem and distribution of primes

Number Theory

Prime Number Theorem and distribution of primes

Prime Number Theorem

I Define π(x) as the function which counts the number of primes up to x.I (One version of) the Prime Number Theorem states:

limx→∞

x/ log(x)= 1. (5)

I Gauss gaussed it (pun intended).I Heuristically, since about x/ log(x) of the x positive integers less than or

equal to x are prime, the “probability” of one of them being prime is about1/ log(x).

Number Theory

Prime Number Theorem

I Define π(x) as the function which counts the number of primes up to x.I (One version of) the Prime Number Theorem states:

limx→∞

x/ log(x)= 1. (5)

I Gauss gaussed it (pun intended).I Heuristically, since about x/ log(x) of the x positive integers less than or

equal to x are prime, the “probability” of one of them being prime is about1/ log(x).

Number Theory

Statistics of primes

I Harald Cramer (1893-1985) made some significant contributions/conjecturestowards understanding the distribution of primes.

I “We are interested in the distribution of a given sequence S of integers, wethen consider S as a member of an infinite class C of sequences, which maybe concretely interpreted as the possible realizations of some game ofchance. It is then in many cases possible to prove that, with a probability 1,a certain relation R holds in C, i.e. that in a definite mathematical sense‘almost all’ sequences of C satisfy R”. 1

I This “pseudo-randomness” gives us some heuristic evidence that someconjectures are true.

I Cramer’s Conjecture:

pn+1 − pn = O((log pn)2). (6)

1“Of course we cannot in general conclude that R holds for the particular sequence S, butresults suggested in this way may sometimes afterwards be rigorously proved by other methods”.

Number Theory

pn+1 − pn = O((log pn)2). (6)

Number Theory

pn+1 − pn = O((log pn)2). (6)

Number Theory

Poisson distribution in primes

I Under Cramer’s idea, we can imagine drawing 1 black/white ball every timefrom an infinite series urns, with probability of a white ball in urn Un being1/ log(n).

I Definezn := I(n-th urn gives a white ball), (7)

Π(x) :=∑n≤x

zn. (8)

I Under Cramer’s framework, Π(x) is a random variable analogy of π(x).I Now, if n is close to some x, then we can think ofzn ∼ Binomial(1, 1/ log(x)).

I For fixed λ > 0, k ∈ N,

#{integers x ≤ X : Π(x+ λ log(x)−Π(x) = k ∼ Pois(λ), (9)

as X →∞.

Number Theory

Π(x) :=∑n≤x

zn. (8)

as X →∞.

Number Theory

Π(x) :=∑n≤x

zn. (8)

as X →∞.

Number Theory

Poisson process and primes

I In other words, under this pseudo-random model of the primes, the randomsets “behave” like a Poisson process:

Number Theory

References

I Robert Gallager. Lecture notes on Poisson Process.

I http://phillipmfeldman.org/mathematics/primes.html

I https://www.youtube.com/watch?v=pp06oGD4m00

I Harald Cramer and the distribution of prime numbers A. Granville. 1993.

I The Coupon Collector’s Problem, M. Ferrante, M. Saltalamacchia, (2014)

I Introduction to Probability Models. S.Ross.

I A First Course in Probability. S. Ross.

I STAT3911: Stochastic Processes Lecture Notes. R. Kawaii.

I https://primes.utm.edu/howmany.html

I https://terrytao.wordpress.com/tag/cramers-random-model/

Poisson Process, Pokémons, and Prime Numbers

Documents

Transcript of Poisson Process, Pokémons, and Prime Numbers