blue wrote:denis_berthier wrote:- a random variable is defined on some measurable space (here the discrete P(M)) with values in some other space (here R); A is a real random variable on P(M);
Are you suggesting that a random variable is a probablity distribution, and nothing more ?
No sorry, I mixed up all: here, A is a random variable with values in P(M).
In a sense, yes, a rv can be considered as a probability distribution on its space of values. More on this at the end of this post.*
blue wrote:denis_berthier wrote:blue wrote:The notation Pr(A=A) is unfamiliar to me.
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Pr(A=A) is exactly what it reads: the probability that A = A.
The problem I would have with it, is that A is a subset of M, and (bold) A is not -- "how could the two be equal ?"
A random variable X is a function: o in O -> X(o), from some probability space O (representing chance) to some space of values (here P(M)). (Read omega for o and O)
Although it is, mathematically speaking, a function, it is called a random variable and the o variable is generally not written: we write X instead of X(o) - the "chance" variable is implicit.
{A = A} is (by definition) the set of elements o in O such that A(o) = A
and Pr(A = A), a shorthand for Pr({A = A}) is (by definition) the probability of this subset of O.
* When you have a random variable X defined on O, it can be used to transfer the probability on O to a probability on its arrival space, here P(M). So, that's what I meant by saying that a rv is almost the same thing as a probability distribution.
[Edit: in Red Ed's case, it'd be much simpler to start from a probability distribution on P(M)]
