E×ample_. Suppose CX

, ,Y

,),

. - , ,

Hn,Ya

.

) are a random

points in the plane . Suppose X, ,Yi , ✗< , Ya , - iv. , ✗n

,Yn

are i. i. d. N lo , 1) random variables.

What is the probabilitythat none of the points is in the disk of radius 2 centred at

the origin .

Sod.

The squared distance from the origin of the point( Xi , Y ;) is ✗ it Yi .

The questions can be written as askingf.- P 1min (✗ it Yi , . . , ✗it Y? ) > 4)From the special case ② above

, ✗it Y? nX} ,which is

the same as the Exponential (E) distribution .

Therefore,

p(min (✗ it Yi , . .

,✗it Yi > 4)

=P ( ✗it Yi > 4 , . . . . ,Xi +Yi > 4)

=p ( ✗ itYi > 4)"

= ¢-4k)"

= e-2h

.

Betadistribu.to#We say

that ✗ has a Beta distribution with parametersa > o

and B > 0,written Beta la , B) ,

if ✗ has pdf

f- (Sc ) ={¥BJ X" 'll →c) P

- l for Ock < I

0

otherwise

where BH ,B) is called the Betafnnct.io# ,

and defined as

B12,B) = f

'

k" 'll - xp " dx

On homework 4 youshowed that

TKS TLB) = That B) Bla , B)

i.is#=FaYiTpY-So an alternative way

to write the pdf isf- (x) =µ%¥%- K

"'

( i - x)B- 1 for o<x< I

0otherwise

momentsofthebetadistribut.io#-eh.ueThe Kth moment is F- [✗ "]

F- [✗ K) = {"

x" F.LI?-gxa-ici-xP-'dx

I

ktt - I[ , - × )B

- I DX÷÷¥¥¥¥f¥¥÷÷x_=\ since integrand is a

proper Beta ( K+2 ,B) density

=

.IT#I::.iiYr.:::?;;:IIp.h-a+.,- using recursive

property of theGamma function)

=i÷:¥⇒:÷+ñThe first 2 moments are

4=1 ! F- IX ) =¥k⇒ : F- 1×9 =¥¥÷+p)TThen

Var LX ) = F-1×9 - F- [×)'

=

ii.is?iiI:iiE+n=Y:I::+;I+;:-*-i-=*%¥•I%•%-d-a↳-#= ÷É+p

.