## Quick Reference

When, as a result of an experiment, values for two random variables *X* and *Y* are obtained, it is said that there is a bivariate distribution. In the case of a sample, with *n* pairs of values (*x*_{1}, *y*_{1}), (*x*_{2}, *y*_{2}),…, (*x** _{n}*,

*y*

*) being obtained, the methods of correlation and regression may be appropriate. Associated with the experiment is a bivariate probability distribution. In the case when*

_{n}*X*and

*Y*are discrete random variables, the distribution is specified by the joint distribution giving P(

*X*=

*x*

*&*

_{j}*Y*=

*y*

*) for all values of*

_{k}*j*and

*k*.

The marginal distribution of *X* is given by , and the marginal distribution of Y is given by . If *X* and *Y* are independent random variables then . The expected values and variances of *X* and *Y* are given in the usual way from these marginal distributions. For example,. The conditional distribution of *X*, given that *Y*=*y** _{k}*, is given by , and the conditional expectation of

*X*, given that

*Y*=

*y*

*, written as E(*

_{k}*X*|

*Y*=

*y*

*), is defined in the usual way as . The conditional distribution of*

_{k}*Y*and the conditional expectation of

*Y*, given that

*X*=

*x*

*, are defined similarly.*

_{j}In the case when *X* and *Y* are continuous random variables, the distribution is specified by the joint probability density function f(*x*, *y*) with the property that if *R* is any region of the (*x*, *y*) plane then . The marginal distribution of *X* then has probability density function (pdf) and the marginal distribution of *Y* has pdf . If the two random variables are independent of one another then f(*x*, *y*) is the product of the pdfs of the two marginal distributions. The expected values and variances of *X* and *Y* are given in the usual way from these marginal distributions.

The probability density function for the conditional distribution of *X*, given that *Y*=*y** _{k}*, is and . See also multivariate distribution.

*Subjects:*
Probability and Statistics.