probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as 2/3. Thus the marginal probability distribution for gives 's probabilities unconditional on , in a margin of the table. Coin flips Consider the flip of two fair coins; let and 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 (unconditional) density functions are The joint probability mass function of and defines 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 die Consider the roll of a fair die 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 joint distribution of and , expressed as a probability mass function, is These probabilities necessarily sum to 1, since the probability of some combination of and occurring is 1. Marginal probability distribution If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individually. The individual probability distribution of a random variable is referred to as its marginal probability distribution. In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables. If the joint probability density function of random variable X and Y is , the marginal probability density function of X and Y, which defines the marginal distribution, is given by: where the first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y. Joint cumulative distribution function For a pair of random variables , the joint cumulative distribution function (CDF) is given by where the right-hand side represents the probability that the random variable takes