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

And

Let the corresponding joint density be denoted by f (R, r).
(a) Using Mathematica or Maple plot this joint density.
(b) Find a function ξ(R, r) such that the interest rates have zero mean under the probability:
dQ = ξ(R, r) f(R, r) dRdr
(c) Plot the ξ(R, r) and the new density.
(d) Has the variance-covariance matrix of interest rate vector changed?

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0.5 0.1 0.1 0.9 μ = | 0.07 0.05