Search results
Results from the WOW.Com Content Network
The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y: []:= [()]. By the Doob–Dynkin lemma, there exists a function : such that
Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of ...
A more general definition can be given in terms of conditional expectation. Consider a function : [,] satisfying (()) = [] for almost all . Then the conditional probability distribution is given by
If the conditional distribution of given is a continuous distribution, then its probability density function is known as the conditional density function. [1] The properties of a conditional distribution, such as the moments , are often referred to by corresponding names such as the conditional mean and conditional variance .
In words: the variance of Y is the sum of the expected conditional variance of Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.
The conditional probability of A given X can thus be treated as a random variable Y with outcomes in the interval [,]. From the law of total probability , its expected value is equal to the unconditional probability of A .
Note: The conditional expected values E( X | Z) and E( Y | Z) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z) is g(Z). Similar comments apply to the conditional covariance.
Taking the expectation of this conditional variance across all values of X gives [ ()], often termed the “unexplained” or within-group part. The variance of the conditional mean, Var ( E [ Y ∣ X ] ) {\displaystyle \operatorname {Var} (\operatorname {E} [Y\mid X])} , measures how much these conditional means differ (i.e. the ...