Jointly distributed Randomvariables
Multivariate distributions
Discrete Random Variables
The joint probability function;
p(x,y) = P[X = x, Y = y]
Continuous Random Variables
Definition: Two random variable are said to havejoint probability density function f(x,y) if
If
then
defines a surface over the x – y plane
Multiple Integration
A
A
f(x,y)
If the region A = {(x,y)| a ≤ x ≤ b, c ≤ y ≤ d} is arectangular region with sides parallel to thecoordinate axes:
x
y
d
c
a
b
A
f(x,y)
Then
A
f(x,y)
To evaluate
Then evaluate the outer integral
First evaluate the inner integral
x
y
d
c
a
b
y
f(x,y)
= area under surface above theline where y is constant
dy
Infinitesimal volume undersurface above the line wherey is constant
A
f(x,y)
The same quantity can be calculated by integratingfirst with respect to y, than x.
Then evaluate the outer integral
First evaluate the inner integral
x
y
d
c
a
b
x
f(x,y)
= area under surface above theline where x is constant
dx
Infinitesimal volume undersurface above the line wherex is constant
f(x,y)
Example: Compute
Now
f(x,y)
The same quantity can be computed by reversingthe order of integration
Integration over non rectangularregions
Suppose the region A is defined as follows
A = {(x,y)| a(y) ≤ x ≤ b(y), c ≤ y ≤ d}
x
y
d
c
a(y)
b(y)
A
Then
If the region A is defined as follows
A = {(x,y)| a ≤ x ≤ b, c(x) ≤ y ≤ d(x) }
x
y
b
a
d(x)
c(x)
A
Then
In general the region A can be partitioned intoregions of either type
x
y
A1
A3
A4
A2
A
f(x,y)
Example:
Compute the volume under f(x,y) = x2y + xy3 over theregion A = {(x,y)| x + y ≤ 1, 0 ≤ x, 0 ≤ y}
x
y
x + y = 1
(1, 0)
(0, 1)
f(x,y)
Integrating first with respect to x than y
x
y
x + y = 1
(1, 0)
(0, 1)
(0, y)
(1 - y, y)
A
and
Now integrating first with respect to y than x
x
y
x + y = 1
(1, 0)
(0, 1)
(x, 0)
(x, 1 – x )
A
Hence
Continuous Random Variables
Definition: Two random variable are said to havejoint probability density function f(x,y) if
Definition: Let X and Y denote two randomvariables with joint probability density functionf(x,y) then
the marginal density of X is
the marginal density of Y is
Definition: Let X and Y denote two randomvariables with joint probability density functionf(x,y) and marginal densities fX(x), fY(y) then
the conditional density of Y given X = x
conditional density of X given Y = y
The bivariate Normal distribution
Let
where
This distribution is called the bivariateNormal distribution.
The parameters are 1, 2 , 1, 2 and
Surface Plots of the bivariateNormal distribution
Note:
is constant when
is constant.
This is true when x1, x2 lie on an ellipsecentered at 1, 2 .
Marginal and Conditionaldistributions
Marginal distributions for the Bivariate Normaldistribution
Recall the definition of marginal distributionsfor continuous random variables:
and
It can be shown that in the case of the bivariatenormal distribution the marginal distribution of xiis Normal with mean i and standard deviation i.
The marginal distributions of x2 is
where
Proof:
Now:
Hence
Also
and
Finally
and
Summarizing
where
and
Thus
Thus the marginal distribution of x2 is Normalwith mean 2 and standard deviation 2.
Similarly the marginal distribution of x1 is Normalwith mean 1 and standard deviation 1.
Conditional distributions for the Bivariate Normaldistribution
Recall the definition of conditional distributionsfor continuous random variables:
and
It can be shown that in the case of the bivariatenormal distribution the conditional distribution ofxi given xj is Normal with:
and
mean
standard deviation
Proof
where
and
Hence
Thus the conditional distribution of x2 given x1 is Normalwith:
and
mean
standard deviation
Bivariate Normal Distribution with marginaldistributions
Bivariate Normal Distribution withconditional distribution
(1, 2)
x2
x1
Regression
Regression to themean
Major axis ofellipses