MATH 3281

Tutorial 1

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.