Editing Joint probability distribution (section)

==Additional properties==
===Joint distribution for independent variables===
In general two random variables <math>X</math> and <math>Y</math> are [[statistical independence|independent]] if and only if the joint cumulative distribution function satisfies
:<math> F_{X,Y}(x,y) = F_X(x) \cdot F_Y(y) </math>

Two discrete random variables <math>X</math> and <math>Y</math> are independent if and only if the joint probability mass function satisfies
:<math> P(X = x \ \mbox{and} \ Y = y ) = P( X = x) \cdot P( Y = y) </math>
for all <math>x</math> and <math>y</math>.

While the number of independent random events grows, the related joint probability value decreases rapidly to zero, according to a negative exponential law.

Similarly, two absolutely continuous random variables are independent if and only if
:<math> f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) </math>
for all <math>x</math> and <math>y</math>. This means that acquiring any information about the value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional (marginal) distribution; thus no variable provides any information about any other variable.

===Joint distribution for conditionally dependent variables===
If a subset <math>A</math> of the variables <math>X_1,\cdots,X_n</math> is [[conditional dependence|conditionally dependent]] given another subset <math>B</math> of these variables, then the probability mass function of the joint distribution is <math>\mathrm{P}(X_1,\ldots,X_n)</math>.  <math>\mathrm{P}(X_1,\ldots,X_n)</math> is equal to <math>P(B)\cdot P(A\mid B)</math>. Therefore, it can be efficiently represented by the lower-dimensional probability distributions <math>P(B)</math> and <math>P(A\mid B)</math>. Such conditional independence relations can be represented with a [[Bayesian network]] or [[Copula (probability theory)|copula functions]].

=== Covariance ===
{{Main|Covariance}}

When two or more random variables are defined on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance. Covariance is a measure of linear relationship between the random variables. If the relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship, which means, it does not relate the correlation between two variables.

The covariance between the random variables <math>X</math> and <math>Y</math> is<ref>{{Cite book|title=Applied statistics and probability for engineers|last=Montgomery, Douglas C.|others=Runger, George C.|isbn=978-1-118-53971-2|edition=Sixth|location=Hoboken, NJ|oclc=861273897|date = 19 November 2013}}</ref>
:<math>\operatorname{cov}(X,Y) = \sigma_{XY}=E[(X-\mu_x)(Y-\mu_y)]=E(XY)-\mu_x\mu_y.</math>

=== Correlation ===
{{Main|Correlation}}

There is another measure of the relationship between two random variables that is often easier to interpret than the covariance.

The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρ<sub>XY</sub> is near +1 (or −1). If ρ<sub>XY</sub> equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight line. Two random variables with nonzero correlation are said to be correlated. Similar to covariance, the correlation is a measure of the linear relationship between random variables.

The correlation coefficient between the random variables <math>X</math> and <math>Y</math> is
:<math>\rho_{XY}=\frac{\operatorname{cov}(X,Y)}{\sqrt{V(X)V(Y)}}=\frac{\sigma_{XY}}{\sigma_X\sigma_Y}.</math>