contains large number blue an urn a green balls The ...stat353/lectures/Examples/Example… · ②...
Transcript of contains large number blue an urn a green balls The ...stat353/lectures/Examples/Example… · ②...
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) .