Regular conditional probability: Difference between revisions

Revision as of 04:30, 31 March 2020

Revision as of 04:33, 31 March 2020

Line 21:

:<math>P(A) = \int_E \nu(x,A) \,P\big(T^{-1}(d x)\big)</math>,

where <math>\nu(x, A)</math> can be denoted, using more familiar terms <math>P(A\ |\ T=x)</math>

(whichthis is "defined" to be conditional probability of <math>A</math> given <math>x</math>, where thiswhich

conditional probability can be undefined in elementary constructions of conditional probability).

As can be seen from the integral above, the value of <math>\nu</math> for points ''x'' outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of ''T''.