> Let F(x) = Integral [-infinite to x] f(u)du be an absolutely

Hi,

> continous distribution function. Prove that corresponding

> probability measure m is given by:

you have f:R->[0,+oo[ measurable such that int f(u)du<+oo. You define

F(x)=int_oo^x f(u)du and you call m the measure on R such that:

m(]-oo,x])=F(x) for all x in R (the fact that such measure exists is

assumed to be granted) . You want to prove that for all B in B(R) you

have m(B)=int_B f(u)du. Define n(B)=int_B f(u)du. It is not difficult

to show that n(.) is a finite measure on R, and you want to prove

that m=n. Define:

D={B in B(R), n(B)=m(B)}

and:

C={]-oo,x]: x in R}

Then C is a subset of D which is closed under finite intersection. It

is not difficult to show that D is a dynkin sytem on R (see

definition on Tutorial 1 of probability.net, it is important to use

the fact that the mesures m and n are finite). From the dynkin system

theorem (see Tutorial 1), the sigma algebra s(C) on R generated by C

is a subset of D. However, s(C) is nothing but B(R) (exercise). So B

(R) is a subset of D and finally D=B(R). This proves that m=n, and we

are done.

Regards. Noel.> > $6 bet on #1 would return $7 if he won, 0 otherwise.

Hi M.W.

> > $2 bet on #2 would return $9 if he won, 0 otherwise.

> > $1 bet on #3 would return $51 if he won, 0

> > otherwise.

you can view the economy as having one time period and three possible

states of the world in the future. State 1: horse 1 wins. State 2:

horse 2 wins etc... A triplet (a,b,c) can represent a financial asset:

you receive a in state 1

you receive b in state 2

you receive c in state 3

Betting on horse 1, is just like buying the asset (7,0,0), the price

of which is 6. The price of (0,9,0) is 2. The price of (0,0,51) is 1.

It follows that the price of (1,0,0) is 6/7, that of (0,1,0) is 2/9

and that of (0,0,1) is 1/51. These (I believe) is what we call the

risk neutral probabilities. Indeed the price of (a,b,c) is (by

linearity):

Price(a,b,c)=a*6/7+b*2/9+c*1/51

So formally, 6/7, 2/9 and 1/51 look like probabilities and Price

(a,b,c) is the expected payoff with respect to these probabilitities.

Now in practice, the bookmaker will sell (1,0,0), (0,1,0) and (0,0,1)

to different people, so as to be as unexposed as possible: so the

bookmaker is trying to sell (1,1,1). Their liability is 1 whatever

happens. However, they have received:

6/7+2/9+1/51=1.09897 ...

So the bookmaker is making a profit of about 10% on its turnover...

Hope this helps.

Regards. Noel