In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases, the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. When both X and Y 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 Y given X 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. Conditional discrete distributions For discrete random variables, the conditional probability mass function of Y given X x can be written according to its definition as: is a version of the conditional expectation of the indicator function for A: An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable. In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulative distribution function, or by a multivariate probability density function together with a multivariate probability mass function. In the special case of continuous random variables, it is sufficient to consider probability density functions, and in the case of discrete random variables, it is sufficient to consider probability mass functions. Examples Draw from an urn Suppose each of two urns contains twice as many red balls as blue balls, and no others, and suppose one ball is randomly selected from each urn, with the two draws independent of each other. Let A and B be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is 2/3, and the probability of drawing a blue ball is 1/3. The joint probability distribution is presented in the following table: Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. In any one cell, the probability of a particular combination occurring is the product of the probability of the specified result for A and the probability of the specified result for B. The probabilities in these four cells sum to 1, as is always true for probability distributions. Moreover, the final row and the final column give the marginal probability distribution for A and the marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, like 2/3. Thus the marginal probability distribution for A gives A's probabilities unconditional on B, in a margin of the table. Coin flips Consider the flip of two fair coins; let A and B be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is 1/2, so the marginal density functions are The joint probability mass function of A and B defining probabilities for each pair of outcomes. All possible outcomes are,,,. Since each outcome is equally likely the joint probability mass function becomes Since the coin flips are independent, the joint probability mass function is the product of the marginals: Rolling a dice Consider the role of a fair and let A 1 if the number is even and A 0 otherwise. Furthermore, let B 1 if the number is prime and B 0 otherwise. Important named distributions Named joint distributions that arise frequently in statistics include the multivariate normal distribution, the multivariate stable distribution, the multinomial distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution. See also Bayesian programming Chow–Liu tree Conditional probability Copula Disintegration theorem Multivariate statistics Statistical interference Pairwise independent distribution References External links A modern introduction to probability and statistics: understanding why and how. Dekking, Michel, 1946-. London: Springer. 2005.. OCLC 262680588.
No comments:
Post a Comment