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In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is ...
A is assumed to be the set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the quotient of the probability of the joint intersection of events A and B, that is, (), the probability at which A and B occur together, and the probability of B: [2 ...
hide. 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 ...
Conditional probability table. In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the ...
Formally, a regular conditional probability is defined as a function called a "transition probability", where: For every , is a probability measure on . Thus we provide one measure for each . where is the pushforward measure of the distribution of the random element , i.e. the support of the . Specifically, if we take , then , and so.
Conditioning (probability) Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory.
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.
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...