** Course outline**:

- Multivariate Distributions.
- The Multivariate Normal Distribution.
- The 1 sample problem.
- Paired comparisons.
- Repeated measures: 1 sample.
- One way MANOVA.
- Two way MANOVA.
- Profile Analysis.

- Multivariate Multiple Regression.
- Discriminant Analysis.
- Clustering.
- Principal Components.
- Factor analysis.
- Canonical Correlations.

Basic structure of typical multivariate data set:

Case by variables: data in matrix. Each row is a case, each column is a variable.

Example: Fisher's iris data: 5 rows of 150 by 5 matrix:

Case | Sepal | Sepal | Petal | Petal | |

# | Variety | Length | Width | Length | Width |

1 | Setosa | 5.1 | 3.5 | 1.4 | 0.2 |

2 | Setosa | 4.9 | 3.0 | 1.4 | 0.2 |

&vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; |

51 | Versicolor | 7.0 | 3.2 | 4.7 | 1.4 |

&vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; | &vellip#vdots; |

** Vector valued random variable**: function
such that,
writing
,

** Cumulative Distribution Function ** (CDF) of : function on
defined by

** Defn**: Distribution of rv is ** absolutely continuous**
if there is a function such that

for any (Borel) set . This is a dimensional integral in general. Equivalently

** Defn**: Any satisfying () is a ** density** of .

For most is differentiable at and

Basic tactic: specify density of

Tools: marginal densities, conditional densities, independence, transformation.

** Marginalization**: Simplest multivariate problem

is the ** marginal** density of
and
the ** joint** density of but
they are both just densities.
``Marginal'' just to
distinguish from the joint density of .

** Def'n**: Events and are independent if

** Def'n**: ,
are **
independent** if

** Def'n**: and are
** independent** if

** Def'n**: Rvs
** independent**:

** Theorem**:

- If and are independent with joint density
then and have densities and ,
and
- If and independent with marginal densities
and then
has joint density
- If
has density and there exist
and st
for (almost)
**all**then and are independent with densities given by

** Theorem**: If
are independent and
then
are independent.
Moreover,
and
are
independent.

Conditional density of given :

Suppose
with
having density .
** Assume is a one to one (``injective") map,** i.e.,
if and only if .
Find :

Step 1: Solve for in terms of : .

Step 2: Use basic equation:

Equivalent formula inverts the matrix:

** Example**: The density

Solve for in terms of :

so that

argument | |||

It follows that

Next: marginal densities of , ?

Factor as where

Then

so marginal density of is a multiple of . Multiplier makes but in this case

Remark: easy to check .

Thus: have proved original bivariate normal density integrates to 1.

Put . Get

So .

2002-09-25