Joint probability distribution

Given random variables X , Y , … {displaystyle X,Y,ldots } , that are defined on a probability space, the joint probability distribution for X , Y , … {displaystyle X,Y,ldots } is a probability distribution that gives the probability that each of X , Y , … {displaystyle X,Y,ldots } falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution. F X , Y ( x , y ) = P ⁡ ( X ≤ x , Y ≤ y ) {displaystyle F_{X,Y}(x,y)=operatorname {P} (Xleq x,Yleq y)} (Eq.1) F X 1 , … , X N ( x 1 , … , x N ) = P ⁡ ( X 1 ≤ x 1 , … , X N ≤ x n ) {displaystyle F_{X_{1},ldots ,X_{N}}(x_{1},ldots ,x_{N})=operatorname {P} (X_{1}leq x_{1},ldots ,X_{N}leq x_{n})} (Eq.2) p X , Y ( x , y ) = P ( X = x a n d Y = y ) {displaystyle p_{X,Y}(x,y)=mathrm {P} (X=x mathrm {and} Y=y)} (Eq.3) p X 1 , … , X n ( x 1 , … , x n ) = P ( X 1 = x 1 and … and X n = x n ) {displaystyle p_{X_{1},ldots ,X_{n}}(x_{1},ldots ,x_{n})=mathrm {P} (X_{1}=x_{1}{ ext{ and }}dots { ext{ and }}X_{n}=x_{n})} (Eq.4) f X , Y ( x , y ) = ∂ 2 F X , Y ( x , y ) ∂ x ∂ y {displaystyle f_{X,Y}(x,y)={frac {partial ^{2}F_{X,Y}(x,y)}{partial xpartial y}}} (Eq.5) f X 1 , … , X n ( x 1 , … , x n ) = ∂ n F X 1 , … , X n ( x 1 , … , x n ) ∂ x 1 … ∂ x n {displaystyle f_{X_{1},ldots ,X_{n}}(x_{1},ldots ,x_{n})={frac {partial ^{n}F_{X_{1},ldots ,X_{n}}(x_{1},ldots ,x_{n})}{partial x_{1}ldots partial x_{n}}}} (Eq.6) Given random variables X , Y , … {displaystyle X,Y,ldots } , that are defined on a probability space, the joint probability distribution for X , Y , … {displaystyle X,Y,ldots } is a probability distribution that gives the probability that each of X , Y , … {displaystyle X,Y,ldots } falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution. The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables. 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 {displaystyle A} and B {displaystyle 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. We can present the joint probability distribution as the following table: