probability density function Cj pdf ),

Multiple

^{Random Variables}

-

Let Xi_, ^{- .} _> Xn be ⁿ random variables . The

vector X ⁼ ( X _{. .} ^{. .}_> Xn

)

^{T is called}

ra¥

^{. So}

X ^{is a} function from ^{S to R"} , where I , P

)

^{is the}

underlying

probability

^{space i X Cs) = ( X,G) , -.., Xnls))T .}

The

distribution

^{of X is the}

probability

^{measure on IR"}

^induced

by

^{P . If Px denotes the} distribution ^{of X} _,

then

_{for ACIR}^"

,

P_, ^CA

)

^{= PC Es ES ! Xo) EA)}

)

^{. Just as}

^with

random variables

we

usually

^describe the distribution ^of X ^via the

following

non

negative

^{functions on IR}

"

:

① ^Joint distribution function ^Cdf

)

⑦ ^Joint

probability

^{mass function}

^Cjpmf )

, in the case

when

Xi , ^-y Xn ^are all discrete ^.

⑦ ^{To .it}

probability density

^function

Cj pdf

^{) , in}

^the

^{case when}

Xi_, ^. Xn ^{are all} continuous random variables .

For our discussion ^{we will assume , unless otherwise}

specified

^,

that

^{all the} Xi^{' ' are} discrete ^{or all} the _Xi^'s are continuous .

Joint Distribution

Functions

-

Let X ⁼ CX _{, .} . .

, Xn )^{' be a random vector. The}

joint

distribution ^{function Cdf}

)

^{of X is}

defined by

Fx ^{( K) = PC X , E K , , . .., X, E kn}

)

, where ^KECK , , ^..>KITE ^R

"

=p

(

^{{ X, ex,} n {X. SKIN . . . n { X. ski}

)

=p

(

^{{ SES :X, Alex ,}

^}

^{n . . . . n { SES ! Xncs) a- kn}

) )

=p

(

^{f - o, K ,}

^]

^{x C - aka) x . .. x C - o, xn)}

)

( note : to^- any ^{2 sets A and PS} ,

A ^x B ⁼ { ^Ca , b) ^{: a EA , be} B)

,

= PCA_.

)

_,

where

_Ae ^- C^{- -} ok ,It ^{. . . x} C^- o_,xn) ^C IR^" _.

Properties

^{of Jon't df's}

-

① ^{o E Fx (x) El for all KEIR}

"

.

⑦ ^Fx

^{Csc) is continuous from}

the right

ⁱⁿ

^each component

^:

if ya ^{d Ki as h → o ,}

then Ling

^{Fx (ki. .-yki- i. Yn , Kit, , ..,kn )}

= Fx ^(ki, ^..,Ki^-i ,Ki _,Kit, , kn

)

Jointprobabilityhassfu.ch#

If ^{X =} (X _{. .} ...

, Xn) ^{T is a} random vector with Xi ^discrete

for all it_, . . .

, n

, then the distribution of X can be

specified

by

^its

^joint pint

^{. To} say that X ^is discrete means

that there

is ^a finite ^or countable subset ^{of IR"} , called the

support

^of

the

distribution of X

,

and

which

^{we will denote}

by

^Sx _,

^that

satisfies ^{P ( X E}

5×1=1

^{. The}

joint ^pmf

^{of X is defined}

by

p× ^{(x) =P ( X = K}

)

^{for all KEIR" . The}

jpmf

pick^{) is} defined ^{for all KEIR" and}

px

^{Cx) so}

only

^{if KE f .}

Also ,

o a- Px ^(x) ^{E l} and

,c⇐§Px

^{Ck) = I . For and ACR" ,}

P ( X EA

)

^{= E}

^pxck

^{) .}

KE AnSx

To.it

Probability Density

^functions

-

If X ^{= (X} _, , ^. , xn)^{T is a} random ^vector

with

_{Xi continuous}

for ^{all it}, ^{. .}, n _, then _if there exists a function

f×

^{! IR" → Lo,m)}

satisfying

^{PC X EA ) =}

af

^{fxck , . . .-, xn) doc, ... dxn for any ACIR"}

then f*

^{is called a} _joint

probability density

^{function of X' .}

Just ^{as continuous random} variables _{may not} have ^a

pdf

,

a continuous

random

^vector _may ^{not have a}

jpdf

^{. In}

^the

vector _case

this

is not hard ^{to see .}

Exampte

^{Let X , be any continuous}

^random

^{variable .}

Let Xa ^{-- X} ,

' and ^X =CX , , Xi)^{T . Then if}

A ⁼ { ^Ck, ,x,) EIR^{' '}. K ^'- ki

)

^{we have PCXEA ) = I .}

If ^a j

pdf

^{for X did exist , say fx ,}

^then

^{it must}

satisfy

I =P ( _X ^{E A}) ⁼

AS

^{fxlx , skaldic , doc. . But there does not exist any}

such function because ^A has ^{0 area in}

R2

.

More

generally

^,

if X ⁼ (X _,

, ^-.> Xn )^{T is an n -}

dimensional

^continuous

random

vector whose

support

^{is contained in a subset of IR" of dimension m ,}

with

^men _, ^then X cannot have a j

pdf

^.

Example

^: Multivariate

Hypergeometric

Distribution

- -

consider

the following ^experiment

^{. We have}

hi

,

objects

^of

_type

^I

nn

"

objects

^of

^type

^r

Let N ^{= hit . .. thr be} the ^{total number of}

objects

^.

Suppose

we draw without

replacement

ⁿ

objects randomly

^{. The}

underlying probability

space ^is CS

, P)

, where S ^{is the set}

of ^all

possible _samples

^{of size n that we could obtain in}

this

way ^{, and P}

specifies

^that _every

sample

^{in S is}

equally likely

^.

Let Xi ^{= # of}

objects

^of

_type

^{i in the}

sample

^{we draw ,}

i ^=L , ^{. .}, ^{r .}

Let X ^{= ( X , . . .}_, ^Xr

)

^{' . Then X is a random vector and its}

distribution ^is called

the

Multivariate

Hypergeometric distribution with parameter

^{n and his . ., her . To consider the}

^{joint pmf}

of ^{X we} should first consider ^the

possible

^{values of X} , i.^e.

,

the

support ^Sx

^{of X .}

ear

^{= 2 , h , =3 , ha = 4 , N}

'

- 7

4th ^{• • • S× when h =/}

3 ^{• • • • it} h ⁼ 2

z• ^{• •}

•

u h =3

, •

!#

⑨ • • JC _, ^{u n -- 4}

° I ² 3 11 n = 5

" h ⁼ 6

" m ⁼ 7

Note that as the

sample

^{size n increases the}

support

^{of X}

starts

hitting

constraints

imposed _by

^{the numbers} hi of each

type

^of

object

^{that there are in the}

population

^.

In

general

^{, we may write the} constraints ^on Sx ^{as follows :}

£ ⁼

{

^{x = (x, , ..., Kr) E Rr : ki E { 0 , --r, hi) for i =L , . ., r}

and ^X_, ^{t ...} t k_, ^{= n}

}

More

explicitly

^{, we can write Sx as}

Sx ⁼ { ^{K = Ck, , ..., kn) E Rr ! x, t .- ut kn = hi ,}

Max(^O, h ^- CN ^-_hi)) ^{E ki E min ( n , hi}

)

_,

and _Ki is an

integer

^{, i =L , . ., r}

^}

^,

Jo_-,it

pmf

Fo^{- ke} Sx _, ^{( K = CK. . . .>}KIT

)

p ( X ^{= x}

)

^{=P ( X, - K, , ---, Xr = Kr}

)

This ^{is a}

counting ^problem

^{. Since all}

^samples

^{in 5 are}

equally

likely

^{, the above}

probability

^is

# of

samples

^{in S that have K, type 1}

objects

, ^..., Kr

type

^r

obj

^.

-

→ total # of

samples

^{in S .}

Note that

samples

^{in 5 are distinct if the}

objects

^{in the}

sample

are distinct _, ^i.e._, ²

samples

^{are distinct if the}

objects

^{in 1}

sample

^{are not all the same as the}

^objects

^{in the}

^other sample

( ^{even if}

they

^{are of the same}

type ⁾

^{. We have}

PCX

, =x

. . ^{. .}, X. ^{= xn}

)

⁼

can ,

So the

joint pmf

^{of X is}

p× ^Cx) ⁼

(74)__.lh

^{if K -- Ck, . ..}

_,kn5

^{' E}

_Sx

{

^O

⁽

^'

^{Ii ) otherwise}

Alternative

Description

^of

^the

Multivariate

Hypergeometric

^Dist'n

-

observe that ^{if X = ( Xi ,. .}_, Xr) ^{T has a} multivariate

hypergeometric

distribution

with parameters

^{n , ni , . .> hr}

then

^the

components

^{of X have the constraint that}

X , t ^... . t X ,

= n

, where n is the _size of the

random sample

^.

Then we

get

P ( ^X _, ^{= k} _{, .} . .

, Xr^-i

-

- kn^- i

)

^=P

⁽

^{Xi = '4 . . .-, Xr-i - Kr-i , Xin -Xi. . -Kr.}

)

^.

=

( ^%)

^{. ..}

^III. Hn

^-

^II

^{. .- xn}

)

That is ^{we can}

equivalently

^describe

^this distribution

as a

distribution ^{on C} Xi . ^{. .}, Xr^-_i

)

^T . The

joint pmf

^of

XXX_, ^{. . . .}_, ^{Xr .}.

)

^{T is}

pxlx ^{. . . .},xr^..

)

⁼

thx _: )

^{. ..}

^{( % :} ) ⁽

^{n -}

^ng

^{. . . - x. ..}

)

when ^we describe

the

distribution ⁱⁿ

this

way ^the support

(

possible

^{values of x, , . ., Xr-i}

⁾

^becomes

5.

×

=

{

^{(Xi , .., Kr..) E IR"' : O E kit . .. t kn, Eh ,}

max

(

^o _, ^{h - ( N -ni})

)

^{E ki E m in Ch , hi}

⁾

and _{Ki is an}

integer

^{, i = I . .., r -I}

}

Alternatively

^{, we can write the}

support

^as

Sx ⁼

{

^{( K, , . .}_, ^{Kru ,) E Rr-t ! O E K , t . ..t kn-i E n}

ki ^E { ^o, ^{. ..}, hi}

,

i ⁼ I_{, .}

..

, r^- I

max ( o _, n ^- CN ^- hi)) ^E _ki

, i = I_, ^. .

, ^r- I

}

Marginal

Distributions ^{( Discrete case}

)

#

Marginal joint ^pmf

^'s

# ^>

Let X ⁼ ( X _{, .} ^..

, Xn

)

^{T be a discrete}

^random

^vector

^with

j ^{o , it}

pmf Px

^{( ki . . . ., kn}

)

^{. Let}

Li

, . ^.., id

)

^{C { Is ..., n}

^}

and

_{{ ji} , ^.. n ^-d

}

^{= { I , . -s n}}

^{) Iii}

^{. . ., id}

^}

^{. We are}

^interested

in the

joint pmf

^{of ( Xi. . . .,}

^{Xia )}

^{" , and in}

^particular

_e ^how

to

obtain

^it from

the

full

joint pmf pick

^{, , . ., xn) .}

^The joint pmf

^{of (Xii , . .}, Xi_,

)

^{T is called}

^the marginal

pmt

^and

^the

distribution of ( _Xi

. . ^{. ..},

Xia)

marginalisation

^{of ( Xi. . ...}, Xia

)

^'

. The _term

marginal

^{refers to} the situation where we are

considering

the joint

distribution of a set ^of random _variables

that

is

a subset of ^a

larger

^{collection of random}

^variables

^.

The

marginal joint pmf

^{of ( Xi . . - r,}

_{Xia )}

^"

_gives probabilities

of

the

_form

P ( _Xi

,

= Ki_, , ^...,

Xia

^- kid

)

^=P

⁽

^{Xi. - Xi , s ..,}

^Xia

^{- kid ,}

Xj

,ER

, ^..,

Xjn.IR )

=

{

Px ⁽Y^{, . . ., Yn}

)

(_Yi,^.._,yn) Est ^{! I}

Yi_, ^-_Ki

, s ^..., Yid ^--_kid

where

I

^{= {Cx, , ...,kn) EIR" :}

pick

^{.. ...,xD >o}

}

is the

support

^{of X}

Example-Margina.to/-theMultivariateHypergeometr#

Suppose

^{{ ii. ..., id}

}

^{C { ' s --i r} and consider the}

joint

distribution of ( _Xi

. . ^. ,

Xia )

^T _. ^The

simplest

_way ^to

_get this

marginal _joint

distribution is to go back ^to the _experiment

giving

^{rise to the} Multivariate

Hypergeometric

distribution and relabel ^all

objects

^{that are not of}

type

^{in , -., id as}

type

^{O .}

Then ^we have hi

,

gbjects

^of

^type

^i,

Nia

"

objects

^of

_type

^id

N ^- ni_,^{- . ..-} hid

objects

^of

_type

^O

and

then

,

letting

^{Xo denote the number of}

objects

^of

_type

^{o in our}

_sample

,

P

(

_Xi, ^{= Ki} _{, ,} ^{. ..}

, Xia ^- _Kid

)

=P (

_Xi

,

- Xi_, s ^- →

Xia

^-- _{did ,} Xo ^{= n - ki. - - n-}

Kid )

=

cail

^{. ..}

^{iii. KI}

^:

^{: :} :

^:

^: Ii :)

-

④

for K ⁼ (Kii ^{, ..}_, kid ) ^e

Sx

^C

Rd

, where

Sx

⁼

{ (

^ki_, ^{. ..},kid

)

^E

Nd

^{! O E Xi}, t ^- - tkid ^{E n}

ma x (o

, n ^- ( N ^- _ni

.

) )

^{E Xin Emin(h , nie}

⁾

and kin ^{is an}

integer

^{, for K = I, . , d}

}