The long rate R and the short rate r are known to have a jointly normal distribution with variance-covariance matrix ? and mean ?. These moments are given by

Let the corresponding joint density be denoted by f (R,r).

(a) Using Mathematica or Maple plot this joint density.

(b) Find a function ?(R,r) such that the interest rates have zero mean under the probability:

dp =??(R,r)f(R,r)dRdr.

(c) Plot ?(R,r) and the new density.

(d) Has the variance-covariance matrix of interest rate vector changed?

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Σ-13 5.1 1.9 . and 07 μ. .05