Examples-

① Joint pmf problem-

Suppose an urn contains a large number of red ,green ,

and blue

balls . The proportion ofred balls is

µ!> o

green' l 'I > O

" with pitpstp, =/a , i ,

blue " " Pz 70 ,we draw balls from the urn with replacement .

Let N ,= # of balls drawn until a red ball is drawn .

iv. * : : : : %: : : : :Hz = #

ca) what is the distribution of Ni ,i =L , 2,3 ?

In.

Ni has a Geometric ( pi ) ,i = I

, 2,3 ,

(b) what is the support of the joint distribution of CN . , Nz, Nz)'?

5ol The support of the distribution of CN, ,Nz

, Nz)'

is

5=5,U Sz US , ,

where

S,= { C l

, j , k ) : j , k are integers 722 and jfk}S,

= { ( i , I,K) : i

,K are integers 22 and i tu. }

S , = { Ci , j , I ) : i, j are integers 22 and i tj }

.

(c) For I < j - k what is PCN ,=/

, Nz =j , Nz -

- K) ?

folk In order for the event IN ,=L, Haj , Nz = K } to occur

,with j - K

,

draw I must be a red ball,draws 2

, . . . , j - I must be red balls,

j th draw must be a greenball

,draws j ti , . . . , K - I must be

red orgreen ,

and draw k must be blue. The probability that such a sequence

of

k draws occurs is p , pi - ' palp , + pz)" - J - '

pz .

So

p ( N ,

= I, Nz

'

- j , Nz -- K ) =p !- '

palp , tpa)" - J - '

Pz,

(d) Compute PCN ,L Nz C Nz ) .

n For { N ,c Na - Nz) to occur we must have N

, =/,and

Nz =j ,N,= k

,for some Kj - hi ,

So we sum upall the

probabilities computed in part (c) .

We have

D

P ( N ,a Nz s Nz) = j.EE PCH ,

=L,Nz --j , Nz -- K)

K -- jt I

=

;.IE?+.p !- '

pipit p . )" -J - '

p ,

=

Ps Pj p !"

. + ,

Cp , +pay- 9 ")

=p , Pa .pi' ' +pj-

1-

=p , I p ,i - i

i"

j =3

=p . ( Ip, - i )= pzl-ci-p.li

-

p ,

= pipa( - pi

② Join't pdf problem-

Suppose (Xi , Xs , Xz)"

is a continuous random vector with joint pdff-×( K , uh , Ks ) =# e

- l"'- "3142

e-(K- "3142

e- kik

←

fo - CK, ,kz.kz/ElR3

.

(a) Find the joint marginal pdf of CX , , Xz )'

.

Solly.

For fixed x,and Ka

, fx (K , .kz ,Xz) as a function of Kz

is a normal pdf up to the correct normalizing constant . The

exponent is a quadratic function of Ks .

We want to complete the

squarein the exponent to write this normal pdf . Writing the exponenent,

- CHIBI - lkzz -

= - I [sci - 2x .sc?tx5txi-zxaKztx5tK5)=- I [ 3×5 - 2x, CK , txz) txt txt)

=- I [ sci - ax, ";# + kit)I

=-Effie, - Kitty - Hita +xit

So, f. ( x . .K

, .sc , ) = (⇒3

e- Z Cx , -

''

II)'

e

- If"; - Hita)-

Sa the joint marginal pdf of ( X , , X. IT is

f, ,(x

, , Ks) = for f-x (Ki , Ks plz ) d Ks Hyt

= e-If"- exited)

.

f- e -Elks-"Yd×

,

inthis is a Nf"t÷

, f) up tothenormalizing constant , which

= :#⇒'

e

-Htt - 'III) " EE .

= ¥5 e-T ( 3k ,

't ski - ( x ,'t zx.sc, txt ))

= ¥5 e-T (Uf - Ki Ka t ki)

,for CK

, .kz )'EIR'

.

(b) Find the marginal pdf of X , .

Solly Start with the joint pdf of (Xi , X. IT .

f-a( x

, , Ks ) = tf e- T (ki - " '

"z tki )

.

We want to integrate• ✓ er x, from - - to it . Completing the square in the exponent gives- T ( x

,

'- kik txt)

= -Y ( sci - ax. txt)=- Tf x . - EY -¥ txt]

=- T Cx . - ¥5 - I sci

so f. ix. ,K) = LE e- T l"- - ¥)

'

e- ¥ " '

'

.

The.the marginal pdf

of X, is

f. (x, ) = !! f. Ix .

.kz/dkz--ITpe-4ki.f-e -Hk . - E) ' d , . - Eo =- I ⇒ o

'-I

-

This is a N (¥,I) up

to

=#og

the normalizing constant ¥FE- ¥r e

- taxi

=EE e- ITI Xi .

Thus,the marginal distribution. of X , is N ( o , 2) .