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In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
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 .
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
To do this, instead of computing the conditional probability of failure, the algorithm computes the conditional expectation of Q and proceeds accordingly: at each interior node, there is some child whose conditional expectation is at most (at least) the node's conditional expectation; the algorithm moves from the current node to such a child ...
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.
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 ...
In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable.The objective is to find a non-linear relation between a pair of random variables X and Y.