I have seen a similar question before and so I am assuming you want me to answer 8.20 and have posted 8.4 just to show the table. However if you require me to solve 8.4 and 8.5 please comment and do it and edit it into my answer asap.
Now covariance of X,Y in Table of problem 8.4= Cov(X,Y)=
Summation( p(X,Y)*(X-x)(Y-y))
Where x and y are the means of X and Y (Usually they are denoted by
mu x and mu y)
x=E(X)= 0.3*1+0.3*2+0.4*3=2.1
y=E(Y)=0.2*1+0.3*2+0.5*3= 2.3
Cox(X,Y)= 0.1(1-2.1)(1-2.3)+ 0.2(1-2.1)(2-2.3) + 0.1(2-2.1) (1-2.3) + 0.2(2-2.1)(3-2.3) + 0.1(3-2.1)(2-2.3)+0.3(3-2.1)(3-2.3)= 0.143+0.066+0.013-0.014-0.027+0.189= 0.37
Now sigma(standard deviation of X= std(x)= 0.3(1-2.1)2+0.3(2-2.1)2+0.4(3-2.1)2=0.69
sigma (standard deviation of y std(y)= 0.2(1-2.3)2 +0.3(2-2.3)2+ 0.5(3-2.3)2=0.61
Now Correlation(X,Y)= Cov(X,Y)/(std(x)*std(y)= 0.37/(0.69*0.61)=0.879
Do ask for any clarifications if required.
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