Transcript Lesson 10 - La Passerelle

Lesson 10
Relation between two RVs
produced in the same experiment
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.
One RV
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Collection of possible values (ai’s)
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
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
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.
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 won’t 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.
Negatively related RVs
Independent RVs
The mathematical concept of
covariance
• Covariance is the formalisation of the
relationship between 2 RVs
• Correlation is a slight variation on
covariance
Computation of covariance
• If we remember how we computed the
variance of one RV…
• …then we extend this to 2 RVs.