# STAT 333 Lecture 27: STAT 333 Lecture 27 Spring 2016

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29 Jul 2016

School

Department

Course

Professor

Lecture 27

Chb :Poisson Process

36-1 Exponential distribution

Exponential distribution :continuous waiting time

.Summary of exp (D)

*,

li )pdf :fcx )={Re X>°

0Otherwise Xis rate

(2) tail pnb :p(x>t)= et 't for t>o

(3) ECX )=¥ &Varma at

(4) No -memory property :

pl Xsttslx >s )=p (X>t )

Meaning : as long as We do notobserve the event ,

Remaining time ~exp (D)

(5) Alarm clock lemma

If Xi ~ expcxi )&Xi ,

";Xn are independent ,

isl ,

-.'in

then (a) min (X,

"

'

,Xn )~e*p( ¥,di )

lb )p( Xi =mih( Xi ,

,.

.

,Xn ))=z±

tXK

Meaning :(a) Xii waiting time for ith type event . it ,

";n

min (Xi ,

..;Xn ):Waiting time for the 1st element

~exp (FE

,xi )

(b) Xi 's min (Xiii ,Xn ): 5th event is the 1st observed event .

&pnob of Xi = min (X,

''iXn )

=

di

=

rate for type i

Sum of rates sum of rates