- Thank you for not only addressing my question, but also clearing many

of the other (mis)conceptions about elementary probability theory.

One of the difficulties I've found on the subject is notations are

highly overloaded with meanings. I'm in the middle of transition

migrating my previous elementary probability theory knowledge to

measure-theoric one and I think your reply clarifies much of my cloud.

> Rmk2: a conditional expectation is always a

Is 1_A the indicator function for event A defined as:

> conditional expectation "knowing some sigma-

> algebra G". For example E[X|Y] is in fact E[X|s(Y)]

> where s(Y) is the sigma algebra on W generated by Y

> i.e. the set {Y^{-1}(B) , B in B(R)}. E[X|Y,Z] is

> in fact E[X|s(X,Y)] where s(X,Y) is the sigma

> algebra on W generated by X and Y. E[X|A] is in

> fact E[X|s(1_A)], where s(1_A)={0,W,A,A^c} etc...

1_A = 1 if A occurs, and 1_A=0 otherwise?

So the second parameter to the expectation notation should be of type

sigma algebra, not of random variable? Right?

> > If

(1) is the exact quote from the textbook. By the way, why using tn

> > P{Y1+...+Yk<tk, k=1...n | Y1+...+Yn-1, tn=u, Y1+...+Yn=y}

> > = 1 - (Y1+...+Yn-1)/tn (1)

>

> For this equation to make sense, I believe you should have

> written.

>

> P{Y1+...+Yk<tk, k=1...n | Y1+...+Yn-1, tn=u, Y1+...+Yn=y}

> = 1 - (Y1+...+Yn-1)/u (1*)

>

> with "u" instead of "tn". So I assume (1*) is the correct

at the RHS doesn't make sense? Remember tn is just an RV.

Thank you again.

Casey > Thank you for not only addressing my question,

Hi Casey, you re very welcome.

> but also clearing many of the other (mis)conceptions

> about elementary probability theory.

> Is 1_A the indicator function for event A defined as:

Strictly speaking yes, the "second argument" to the

> 1_A = 1 if A occurs, and 1_A=0 otherwise?

> So the second parameter to the expectation notation

> should be of type sigma algebra, not of random variable?

> Right?

*conditional* expectation should always be a

sigma-algebra, e.g. E[X|G]. However, as very often

in mathematics, notations get a little bit sloppy,

when there is no ambiguity, when it looks nicer or

when it is more convenient. So if Y is a random variable,

and s(Y) is the sigma-algebra generated by Y, *no one*

will ever write E[X|s(Y)], but E[X|Y] instead. It is

more convenient to write E[X|Y] and there is no risk

in doing so, since everyone knows it means exactly

E[X|s(Y)]... Similarly, if A is some event (an element of F)

everyone will write E[X|A] instead of E[X|1_A] or E[X|s(1_A)]

or E[X|{0,W,A,A^c}]... Also, if B is another event,

everyone will write P(B|A) instead of E[1_B|A]... Note

that strictly speaking P(B|A) is a *random variable* and

not a number. In all elementary textbooks, you will

see the formula:

P(B|A)=P(B/\A)/P(A) (*)

where in fact, strictly speaking, P(B|A) is the random

variable:

P(B|A)=(P(B/\A)/P(A))*1_A + (P(B/\A^c)/P(A^c))*1_A^c

For all outcome w in A, (i.e. such that 1_A(w)=1), we have:

P(B|A)(w)=P(B/\A)/P(A)

I believe equation (*) is damaging because it is imcompatible

with the true definition of E[X|G] (which means that beginners

get very confused I would think)

> at the RHS doesn't make sense? Remember tn is just an RV.

Yes, tn is a random variable. If you write something like

E[X|tn], then this is also a random variable, which is measurable

with respect to s(tn), i.e. which can be expressed as a

(measurable) function of tn, i.e. E[X|tn]=g(tn) for some

measurable function g:R->R. So an equation like E[X|tn]=g(tn)

makes perfect sense. Now, by definition, E[X|tn=u] is just

another notation for g(u). More precisely, E[X|tn=u] is

supposed to refer to the "value at u" of "the" (not quite

unique) function g such that if you take the composition

of g and tn, i.e. (g o tn) you obtain E[X|tn], or in other

words the function g such that E[X|tn]=g(tn). Hence an equation

like E[X|tn=u]=g(u) makes perfect sense. But writing something

like E[X|tn=u]=g(tn) makes no sense in my opinion... Neither would

E[X|tn]=g(u) be meaningful...

Noel.