Relation between two RVs

1
Lesson 10

• Relation between two RVs
• produced in the same experiment

2
Portfolio construction

• We want to spend our money into several
  securities bought in the market.
• So we need to study several securities (and
  therefore RV) together.
• If securities were independent (on a market,
  think of the NYSE, and the dollar) it would be
  easy to build a portfolio with high return and
  zero risk.

3
One RV

• Collection of possible values (ais)
• Set of probabilities
• A long series of outcomes
• A histogram
• The outcomes were produced by replication of an
  experiment E
• In finance, the usual experiment is wait one
  year
• Two concepts the mean and the variance

4
Two RVs (produced in the same experiment)

• All the initial concepts are naturally extended.
• But there will also be a new one.
• Old ones
• The set of possible values a collection of
  pairs (ai, bj)s
• A set of probabilities each pair has a
  probability
• A long series of outcomes we can plot them and
  construct the extension of a histogram a
  scattergram, and we count pairs that fell in each
  cell of a grid

5
2 RVs the new concept

• The new concept is the variables may or may not
  be related
• Some joint distributions (i.e. the set of
  probabilities) show independence, and some set of
  probabilities reveal dependence.
• This can also be seen with scattergrams drawn
  from long series of actual outcomes of pairs.
• Concept introduced by Karl Pearson (working with
  Francis Galton, a cousin of Charles Darwin) in
  the second half of the XIXth century, while
  studying the role of genetics in evolution and
  related topics in biology and agriculture.

6
Visual interpretation

• The best way to feel the relationship between
  two RV is to look at their scattergram (or their
  joint distribution)
• We fit an oval shape (with the appropriate
  technique which we wont study) through the
  scattergram
• The most important fact is whether there is an
  angle between the axes of the oval and the x and
  y axes
• If the angle is zero no relationship between X
  and Y.
• If there is an angle there is a relationship
• The narrowness of the oval is related the
  strength of the relationship.

7
Negatively related RVs
8
Independent RVs
9
The mathematical concept of covariance

• Covariance is the formalisation of the
  relationship between 2 RVs
• Correlation is a slight variation on covariance

10
Computation of covariance

• If we remember how we computed the variance of
  one RV
• then we extend this to 2 RVs.