Chapter 15 - Section 1.3

[PBhella]

I'm struggling to prove equations (1.1) and (1.2) in the course notes.

(1.1)
There are two proofs given on (1.1):
1. E[X] = Int(0,Inf)[x*dF(x)]
2. E[X] = Int(0,Inf)[S(x)dx]

I understand how 1. can be proved. However, for 2. I can't follow the course notes which suggest using Integration by Parts and substitution where u=S(x)=1- F(x) and dv/dx=1.

Could somebody please explain the above on (1.1) and proof behind (1.2)?

Thanks

 

[Darren Michaels]

Hi PBhella

Please see the attached proof of the RHS of equation 1.1.

You should be able to follow a similar approach to prove the RHS of equation 1.2.

Have a go and see how you get on.

 

[PBhella]

Thanks but I don't understand how we get the S(x)*x term in (1.2).

 

[Darren Michaels]

Its exactly the same as the above proof, except that the integration of S(y) is now from 0 to x instead of from 0 to infinity.

The S(x)*x term comes from the first term (when y=x) when you integrate by parts.

 

[PBhella]

Hi, I've revisited (1.2).
The course notes state (1.2) as: E[X^x] = x*S(x) + Int[y:0 to x][y*dF(y)] = Int[y: 0 to x][S(y)*dy]

I understand how to prove the last two equalities. However, I don't understand how they equal E[X^x]. Could somebody please explain?

Thanks

 

[Darren Michaels]

This is simply the definition of LEVx(X). The limited expected value of the random variable X, where values are limited to x (at most).

So the first part of the expression to the right of the equality deals with the values of X>x and the integral when X<=x.