Chapter 6, Problem 6.6P

The severity of a certain cancer is designated by one of the grades 1,2,3,4 with 1 being the least severe and 4 the most severe. If X is the score of an initially diagnosed patient and Y the score of that patient after three months of treatment, hospital data indicates that $p\left(i,j\right)P\left(X=i,Y=j\right)$ is given by $\begin{array}{l}p\left(1,1\right)=.08,p\left(1,2\right)=.06,p\left(1,3\right)=.04,p\left(1,4\right)=.02\\ p\left(2,1\right)=.06,p\left(2,2\right)=.12,p\left(2,3\right)=.08,p\left(2,4\right)=.04\\ p\left(3,1\right)=.03,p\left(3,2\right)=.09,p\left(3,3\right)=.12,p\left(3,4\right)=.06\\ p\left(3,1\right)=.01,p\left(4,2\right)=.03,p\left(4,3\right)=.07,p\left(4,4\right)=.09\end{array}$

a. Find the probability mass functions of X and of Y;

b. Find E[X] and E[Y].

c. Find Var (X) and Var (Y).