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Concept
Conditional probability distribution
Résumé
In probability theory and statistics, 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. When both and are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.
If the conditional distribution of given is a continuous distribution, then its probability density function is known as the conditional density function. 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.
More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables.
For discrete random variables, the conditional probability mass function of given can be written according to its definition as:
Due to the occurrence of in the denominator, this is defined only for non-zero (hence strictly positive)
The relation with the probability distribution of given is:
Consider the roll of a fair and let if the number is even (i.e., 2, 4, or 6) and otherwise. Furthermore, let if the number is prime (i.e., 2, 3, or 5) and otherwise.
Then the unconditional probability that is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that conditional on is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).
Source officielle
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