In article <[EMAIL PROTECTED]>, John Hicken <[EMAIL PROTECTED]> wrote: >*** post for FREE via your newsreader at post.newsfeed.com ***
>I'm doing a module on Probability Theory, and I'm not sure I 'get' the >concept of conditional probability of a random variable over a sigma field. >I think I understand the mathematics of the situation, but I don't really >have an intuitive grasp of the situation. >By a sigma field I mean a set B, of subsets of a set A, containing the empty >set, and closed under taking complements, and countable intersections. >According to the definition I've been given, for a random variable Y on >probability space (A,F,P), with E[|Y|] < infinity and D a sub sigma field of >F. A version of E[Y|D] is any function from A to the real numbers that is D >measurable, and whose integral over each member of D is the same as that of >Y, the integrals being over P. I say all that so we know exactly what we >are talking about. >What seems to be happening is that the expectation over a sigma field D, is >sort of sampling the random variable Y, so it acts in the same way over the >sets in D. This is incorrect. >But perhaps someone could tell me if there is a better way of understanding >intuitively what this means. Since the argument is by means of the Radon-Nikodym Theorem, we need to go back to that to get the understanding. BTW, one can define conditional expectation with respect to a measure which is not a probability measure, and this can be useful. At any rate, suppose we have a finite measure space (A, D, P), and we have a (signed) measure Q on D, and for convenience we will assume that |Q(S)|/P(S)| is bounded for all S in D. We do not need such strong assumptions, but the R-N theorem can be reduced to this, and the intuition remains valid. Now for any partition Z of D into a finite (countable can be used as well) family of sets of positive P measure, define r_Z(x) = Q(S)/P(S) for that S which contains x. Take a sequence of partitions Z_n which spreads out the r's; under the conditions given, making \int (r_Z_n)^2 dP go to its least upper bound will work, but otherwise a maximal spreading out is needed, and any will do. Then r_Z_n will converge in measure, and the limit is the Radon- Nikodym derivative, which for your problem is the conditional expectation. So the intuition is to take conditional expectations on finite partitions, and pass to the limit using a sequence of partitions which use as much of the information as they can. It is the natural extension from discrete partitions. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
