Poisson goes here

Poisson RV parameter x chapter 3.54

counting occurrences in a given Time periodor a given spatial region if events

are completely random in space o timeEx emissions from a radioactive substance

requests for telephone connections arrivals

defects in a chip

PLN K pm Lk 4 ea fr leo 1,3

a average of events occurring in a

specific interval or region

pmf sums to 1 Ey 9 ea

e info 9ELN a e ed IVar N a

Good for modeling when events are rare smallpbut the trials is large large n

Approximates binomial well in these instancesand its much easier to compute than binomialfate

Examples Poisson

number of queries N at a call centerin t seconds is a Poisson RV

with a Xt where X is the

average arrival ratein a period of time

t

Ex X 4 queries minute

note units of are same as units of RV

Q1 find the probability of more than 4 queries

in 10 seconds

Given X find a

48wm.inte fomisneIgna losecmds F8neries

units of RV

p N 4 I PINE 4I 3

k

mo Tei expfI 6.33 lo 4

Makes sense this is small because if the

avg queries in 10 seconds is 43 its

quite unlikely to get more than 4

Q2 what is the probability there are fewer

than 6 queries in 2 minutes

Given the new time duration we have a new x

x fifties 2 minutes 8 queries

5 gk

Pl N E 5 E t exp 8 oil

k O

This also makes sense the average is 8

so it's unlikely there are 5 or less

printedExercise on

Poisson goes here

Another Poisson Example

packets in a network arrive at a node at a rate

of 100 packets per minute What is the probability

no packets arrive in 6 secondswhat is the

probability 2 or more packets arrive in 6seconds

It Pamdifte minutes

10 packets

Pm od

ea

e e a 4.5 10 4

p l N 2 l p N o P N L

l pv o pulll e

to

ay eto

I 0 9995

Poisson approximates binomial fo large h smallp

Define a _hp and let n 00

Then puck Y pk tp I 44 ewhen x np

Proof step1 examine case where no event occursin n trials

polo Li p µ Fn e as nooo

step 2 consider the ratio 1Pu k

f n trials

Eis Kian a iEt Ee iiI ii n

This II as n 00

So as n 00 pulkti I Paulk IT

and so pn k q ea

F Pulo

Isince pulo ea

what is large n How small is smallp

Matlab plots of

approximations

Binomial geometric

comparison

Geometric RV Binomial RVparameter p parameters n p

The number of independent The number of successes

Bernoulli trials until the in an independentfirst success Bernoulli trials

SO

µHip Pgpm

pµ pfy.rs lNl n levelsm 21

µp each branch 8th the

m 3 was the same depth

Random variable XRandom variable M

M E 1,2 3 X E 0,1 2,3 n n

PmLk d pl p p K E pka p

whether you continue the times gon perform

the subexperiments or noteach subexperiment does

depends on the outcomenot depend on the outcome

of a previous subexperiment ofa previous subexperimen