The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i.e., a random variable satisfying E( | X | ) < ∞) and Y is any random variable, not necessarily integrable, on the same probability space, then Probability theory is the mathematical study of probability. ... A random variable is a term used in mathematics and statistics. ... In mathematics, a probability space or probability measure is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...

i.e., the expected value of the conditional expected value of X given Y is the same as the expected value of X. In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

The nomenclature used here parallels the phrase law of total probability. See also law of total variance, variance decomposition. Nomenclature in probability theory is not wholly standard. ... In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then In language perhaps better known to statisticians than to probabilists, the first term is the unexplained component of the variance... The well-known variance decomposition rule is given by: See also iterated expectations and law of total variance for proof. ...

(The conditional expected value E( X | Y ) is a random variable in its own right, whose value depends on the value of Y. Notice that the conditional expected value of X given the event Y = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Y = y) = g(y) then the random variable E( X | Y ) is just g(Y). ) In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. ... In probability theory and statistics, some special forms of mathematical notation are of interest : Random variables (for example, the height of students) are written in upper case. ...