Regular conditional probability: Difference between revisions

Revision as of 12:47, 29 March 2011

Revision as of 12:48, 29 March 2011

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==Example==

To continue with our motivating example above, we consider a Radon space <math> \Omega </math> and a real-valued random variable ''X''. As discussed above, in this case there exists a regular conditional expectation with respect to ''X'' and we may write

:<math>\mathfrak P(A|X=x_0) = \nu(x_0,A) = \lim_{\epsilon\rightarrow 0+} \frac {\mathfrak P(A\cap\{x_0-\epsilon < X < x_0+\epsilon\})}{\mathfrak P(\{x_0-\epsilon < X < x_0+\epsilon\})},</math>

(where <math>x_0=2/3</math> for the example given.) This limit, if it exists, is a regular conditional probability for ''X'', restricted to <math>\mathrm{supp}\,X.</math>