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Lesson 10

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  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 (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

  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 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.

  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.

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