Regular conditional probability: Difference between revisions

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Revision as of 17:37, 8 January 2021

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==Motivation==

Consider a discretetwo random variablevariables ''X'' thatand ''Y'', where the represents the roll of a die.

The conditional probability an eventof ''EY'' being in a Borel set <math>A \subseteq \mathbb{R}</math> is given by

:<math>P(EY \in A | X = x) = \frac{P(Y \in A, X = x)}{P(X=x)}.</math>

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

:<math>\nu(x, EA) = P(EA | 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''.