Regular conditional probability: Difference between revisions

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In [[probability theory]], '''regular conditional probability''' is a concept that formalizes the notion conditioning on the outcome of a [[random variable]]. The resulting '''conditional probability distribution''' is a parametrized family of probability measures called a [[Markov kernel]].

{{short description|Alternative probability measure conditioned on a particular value of a random variable}}

'''Regular conditional probability''' is a concept that has developed to overcome certain difficulties in formally defining [[Conditional probability|conditional probabilities]] for [[continuous probability distribution]]s. It is defined as an alternative [[probability measure]] conditioned on a particular value of a [[random variable]].

==Motivation==

Normally we define the '''conditional probability''' of an event ''A'' given an event ''B'' as:

Consider a discrete random variable ''X'' that represents the roll of a die.

:<math>P(A|B)=\frac{P(A\cap B)}{P(B)}.</math>

NormallyThe we define the '''conditional probability''' of an event ''AE'' is given an event ''B'' as:by

The difficulty with this arises when the event ''B'' is too small to have a non-zero probability. For example, suppose we have a [[random variable]] ''X'' with a [[uniform distribution (continuous)|uniform distribution]] on <math>[0,1],</math> and ''B'' is the event that <math>X=2/3.</math> Clearly, the probability of ''B,'' in this case, is <math>P(B)=0,</math> but nonetheless we would still like to assign meaning to a conditional probability such as <math>P(A|X=2/3).</math> To do so rigorously requires the definition of a regular conditional probability.

:<math>P(AE |B X = x) = \frac{P(A\cap, BX = x)}{P(BX=x)}.</math>

Conditional probability forms a two-variable function <math>\nu:\mathbb{R} \times \mathcal{F} \to \mathbb{R}</math>

:<math>\nu(x, E) = P(E | X =x)</math>

Note that when ''x'' is not a possible outcome of ''X'', the function is undefined: the roll of a die coming up 27 is a probability zero event. The function <math>\nu</math> is defined [[almost everywhere]] in ''x''.

Now consider two continuous random variables, ''X'' and ''Y'', with density <math>f_{X,Y}(x,y)</math>.

The conditional probability of ''Y'' being in a region <math>A \subseteq \mathbb{R}</math> is given by

:<math>P(Y \in A | X = x) = \frac{\int_A f_{X,Y}(x, y) \mathrm{d}y}{\int_\mathbb{R} f_{X,Y}(x, y) \mathrm{d}y}.</math>

Conditional probability is a two variable function as before, undefined outside of the [[support]] of the distribution of ''X''.

==Definition==