Bivariate or joint probability distributions

∑ ∑ , = 1.
Also p( x y)
x
y
2
3.3 Marginal probability distributions
The marginal distributions are the distributions of X and Y considered separately
and model how X and Y vary separately from each other. Suppose the probability
functions of X and Y are p
X ( x)
and pY ( y)
respectively so that
p
X ( x)
= P(X = x) and p ( y)
Also ∑ p
X ( x)
= 1 and pY
( y)
x
∑ = 1.
y
Y
= P(Y = y)
It is quite straightforward to obtain these these from the joint probability distribution
p x p x , y p y p x , y
since
X ( ) = ∑ ( ) and
Y ( ) = ∑ ( )
y
In regression problems we are very interested in conditional probability distributions
such as the conditional distribution of X given Y = y and the conditional distribution
of Y given X = x
x
3.4 Conditional probability distributions
The conditional probability function of X given Y = y is denoted by p( x y)
is defined as
p( x y ) = P( X x Y y)
= = =
( = = )
P( Y = y)
P X x and Y y
=
( , )
( y)
p x y
whereas the conditional probability function of Y given X = x is denoted by p( y x)
and defined as
p( y x ) = P( Y y X x)
= = =
( = = )
P( X = x)
P Y y and X x
=
p
Y
( , )
( x)
p x y
p
X
3.5 Joint probability distribution function
The joint (cumulative) probability distribution function (c.d.f.) is denoted by F(x, y)
and is defined as
F(x, y) = P( X ó x and Y ó y) and 0 ó F(x, y) ó 1
The marginal c.d.f ’s are denoted by FX ( x)
and F ( y)
F
X ( x)
= P( X ó x) and F ( y)
(see Chapter 1, section 1.12 ).
Y
= P( Y ó y)
Y
and are defined as follows