How to estimate a discrete bivariate distribution using a bivariate normal distribution in the following problem

Assume there are two drugs — A and B — to treat a disease. For 20% of people having the disease, A helps but B does not help. For 20%—B helps but A does not help. For 40%—both A and B help, and for 20%— neither of the two drugs helps. Assume that 20 persons having the disease are chosen at random. Let X be the number of people (among those 20) for whom drug A helps, and Y — the number of people for whom drug B helps. The joint probability mass function(p.m.f) of the pair (X, Y ) can be computed as (SEE ATTACHED DOCUMENT).

  • Compute means and variances of X and Y and their correlation coefficient.
  • What is a natural bivariate normal distribution to approximate the joint distribution of the pair (X, Y )? Explicitly write down the joint probability density function(p.d.f) of this bivariate normal (W,Z)
  • Let g denote the joint probability density of (W, Z). Draw the level sets of the joint p.d.f g(t,w) of the bivariate normal (W,Z) (i.e. the sets corresponding to g(t,w) = constant)

Work must be shown. However, online tools such as Wolphram Alpha can be used.