Joint, Marginal, and Conditional Distributions Page 1 of 4
Joint, Marginal, and Conditional Distributions
Problems involving the joint distribution of random variables X and Y use
the pdf of the joint distribution, denoted ).,(
,yxf YX This pdf is usually given,
although some problems only give it up to a constant. The methods for
solving problems involving joint distributions are similar to the methods for
single random variables, except that we work with double integrals and
2-dimensional probability spaces instead of single integrals and
1-dimensional probability spaces. We illustrate these methods by example.
Discrete Case: Analogous to the discrete single random variable case, we
have
1))()Pr((),(0 ,
≤
=
∧
=
=
≤yYxXyxf YX
The Continuous Case is illustrated with examples.
The Mixed Case (one of the random variables is discrete, the other is
continuous) is also illustrated with examples.
cdf of Joint Distribution – denoted ),(
,yxF YX
))()Pr((),(
,yYxXyxF YX
≤
∩
≤
=
Notice that we can get the cdf’s for X and Y from the joint cdf as follows:
),()(
),()(
,
,
yFyF
xFxF
YXY
YXX
∞=
∞
=
