in probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. in other words, it is the expected value of one variable given the value(s) of one or more other variables. it is also known as conditional expected value or conditional mean. the concept of conditional expectation is important in kolmogorov's measure-theoretic definition of probability theory. the concept of conditional probability is defined in terms of conditional expectation. == calculation == let x and y be discrete random variables, then the conditional expectation of x given the event y=y is a function of y over the range of y : \operatorname{e} (x | y=y ) = \sum_{x \in \mathcal{x}} x \ \operatorname{p}(x=x|y=y) = \sum_{x \in \mathcal{x}} x \ \frac{\operatorname{p}(x=x,y=y)}{\operatorname{p}(y=y)}, where \mathcal{x} is the range of x. if now x is a continuous random variable, while y remains a discrete variable, the conditional expectation is: : \operatorname{e} (x | y=y )= \int_{\mathcal{x}} x f_x (x |y=y) dx where f_x (\,\cdot\, |y=y) is the conditional density of x given y=y. a problem arises when y is continuous. in this case, the probability p(y=y)&nbsp;=&nbsp;0, and the borelâkolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines. however the above expression may be rearranged: : \operatorname{e} (x | y=y) \operatorname{p}(y=y) = \sum_{x \in \mathcal{x}} x \ \operatorname{p}(x=x,y=y), and although this is trivial for individual values of y (since both sides are zero), it should hold for any measurable subset b of the domain of y that: : \int_b \operatorname{e} (x | y=y) \operatorname{p}(y=y) \ \operatorname{d}y = \int_b \sum_{x \in \mathcal{x}} x \ \operatorname{p}(x=x,y=y) \ \operatorname{d}y. in fact, this is a sufficient condition to define both conditional expectation and conditional probability. == formal definition == let \scriptstyle (\omega, \mathcal {f}, \operatorname {p} ) be a probability space, with a random variable \scriptstyle x:\omega \to \mathbb{r}^n and a sub-&sigma;-algebra \scriptstyle \mathcal {h} \subseteq \mathcal {f} . then a conditional expectation of x given \scriptstyle \mathcal {h} (denoted as \scriptstyle \operatorname{e}\left[x|\mathcal {h} \right]) is any \scriptstyle \mathcal {h} -measurable function (\omega \to \mathbb{r}^n) which satisfies: : \int_h \operatorname{e}\left[x|\mathcal {h} \right] (\omega) \ \operatorname{d} \operatorname{p}(\omega) = \int_h x(\omega) \ \operatorname{d} \operatorname{p}(\omega) \qquad \text{for each} \quad h \in \mathcal {h} .loÃ¨ve (1978), p. 7 no
Probability_theory Statistical_terminology