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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 .
Here, in the earlier notation for the definition of conditional probability, the conditioning event B is that D 1 + D 2 ≤ 5, and the event A is D 1 = 2. We have () = () = / / =, as seen in the table.
Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter.
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 ...
In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel .
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
The concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible. For we can obtain a probability distribution for [the latitude] on the meridian circle only if we regard this circle as an element of the decomposition of the entire spherical surface onto meridian circles with the given poles
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let (,,) be a probability space.Suppose is a random variable with distribution function , and an event on (,,).