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Tutorial 2. MATH 3221
Proposition
Let JCa) ( :=J ) denote the r.ir . representing thenumber of fractional time units ( a ) lived during
the year of death.
Then it the UDD holds on @ )then Kla ) k JCA ) are 1
.i
proof Let's K :-c Kk ) j Ji=J( u ) ,then
P[K=k,
Jcj ) =P [ KIT < ktj ) =P ( k <TE ktj ]
UDDa kpu jqutie = ( k Pu 9utk ) J
= klqu . j = PCKK )=k ) P[J( a) cj ]
This completes the proof . 6
Note C Recall cfh means ← Pai EH )What if CFM holds on (a) ? Cef 's trg :
P[K=K,
Jcj ] = PL K I T < Ktj ) =P ( Kc T { k←j )
" kpu j9utk =
Kpn ( c - Jputie )
Input( , . ( pun . )s ) . Pu" ( i . Pu) ( i - ( Putu )
')
1- Pu
So ?
Ex 1
If pcx ) =p C- Co ,p ),
what is A- x?
÷a
= § e- ftp.cpcxetldt = § e- rfpe ' ttde
, § p e-C red ' ←
at a hyper )
Ex 2 s A-x ;i an increasing / decreasingfunction in i 7
£.
otxci ) =¥ §e- otpxpcxet ) It
. I,
If1)←
← pxpcxetldt
= § ¥,
( Ii ) ttpxpcxetidt
a { -
'
f ( Eu )←
2k¥ ,)2 tpxpcxet ) at
= § . c- (Eu )'s + 1
← papcxetldt
⇐ - v § Evtcpx pcxetldt a - v ECTG )vtC⇒]
co
Another wag would be to look af
jdg eceot ] = effzeiot ]⇒E(Tent ]
< 0
Ex 2
Show Ex = eat I
Sol
Based or the proposition Tcx ) =kG7+JCa )
k Kcal I Jcx ) n U Co ,D
£x= ECTGD ) = E[ Kcx ) + JG ) ] eat ECJCO . ) )
2 Ex t £ .
Note The Laaf that GDD holds on e. g.( x )
doesn't mean It holds on Cu ).
Pnot
The UDD holds on (a) iff for Kf # + ,te C 0,17
keeper = ( 1-E) kpu + tueipu
Assume UDD holds on C x ),
hence
Ket Px = @-f) * Pxt Eueipx the # ,tE[ 0,2 )
Take (a) := ( x :D ),
check,
# r K=n - 2
net Px :D = PCKCX :D ) > Ket ]
#Recall
Spx,
Scn
Spx:n7 2 { 0,
s > a
hence
@ Lr K=n - I,
we obtain ntetpx :D 2 nuetpx + fflo,
' )
Also nips :a= napa knpx :a =0
Sogenerally ( It ) nipxfntet
Pa ,which is
a contradiction.
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