Theorem Grew that Nltl n the n arrival times S Sn

have the same distribution as the order statistics corresponding

to n independent random variables uniformlydistributed on the

interval at

D let Ya Yn be v.v s Their order statistics

but 7221 Tenn are the values of Yi Tn in increasingorder

Last time we showd that if Ya Tn are i i d UniformCEOED

the joint Pdf of Ya i r Yen is fly yn n for any 0 Lie synst

Proof of Theorem We will show that the jointdensity ofn

Se SngivenNCH nis fGii sn Nltsn n

t s no events

n 3O F EI S3

The event Si si Sn Sn Nlt n for see snoot

is the same as the event that the first n I interarrnal

times T Tnt satisfy Ta Si Tze Sz Si Tn Sn Sn e

Tnt t Sn

Thus fCsi Sn INCH n fCsi sis Tnt t Sn

p n

f LI Si Tz Sz Si Tn Sn Sn e Tum t Sn

FthX e S de des si de Nsu

Snele dit Sn yn e At

City Intent Its

Now suppose thatwe label events of a Poisson process into a

possible types s t at each true s Keene is a distribution Pics

i t k s t PCerent occurring at times is type i Pics

the type selection B indep of what previouslyoccurred

Proposition let Milt Type i events by true t Then N Ith n Nottare independent Poisson v.v s woke means ECNiltB Picsds

Reina We recover part of the Poisson thinning proposition if pi Pthen E Nitti dfotpds p.tt

Proof PCN Lt en Niall the

PCN Ctl n NIH nel Nlt II ni PCNltl i.IN

Condofronal on n events arrival tunes are indep and unifor

on O t so P event is type i J PCerentistypeeil eventoccursat

IIt ftp.cslds pi probof

Each of thee in events has prob Ti of beingtypeiannual

multneoveralat S

and they are independent distribution

PCN Ltl n Nield nul Nlt Ining nI PT

Fah

lPCN IH hi Nuits na

n naPT pine Cdt

t

nd

j'e 4T ftp.T e tE

we used that net the n and ZpT l

N It Nutt are independent and

Nolet n Poisson d ftp.cslds

Exaeuploo Let's get back to our examples individuals independently

getinfected following a d Poisson process s t symptoms appear

after true Tsince infection where T has c De g

N CES individuals infected with symptoms

µ Lt a individuals infected with no symptoms

NCH N Lt tweet total ofindividuals infected

Goyal Fred E Nuts which is an estimate of Walt

Solution Fix t and use previous proposition with k 2

P N Ctl e n Nz t na P Natt n Nz H ha Nlt ni tha

P Nlt n tha

Tor OES Et an individual infected at time s has

symptoms by true t w p GRE S and has no symptoms

by true t wp f Get sM's

PisN CH r Poisson CA ft Gct sidsNzCEI n Poisson L X ft I CtDdg

ECN.HNSotEtEfwdsnIECNzCHJdgt GctsDds

estimate

To estimate A since we wow M LH SaezN CH n number ofeople infected with symptomsby tweet then use

na E EIN Its L Sot Gct ssds to get an estimate

I fatsoes ni Ifta at shoes mffIII.fids

e g t 16 days G n Eep daygn 2000 then viz 2000 Joke ds

eF dJ

2000 OCI e16 10

655 i Iz 2114