5. You roll a pair of fair dice independently. What is the correlation coefficient of the...

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5. You roll a pair of fair dice independently. What is the correlation coefficient of the high and low points rolled? Hints:

math Statistics-And-Probability

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Answer 1

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X is the low points rolled and Y is the high points rolled.. Then the joint p.m.f is

X Y	1	2	3	4	5	6
1	1/36	0	0	0	0	0
2	2/36	1/36	0	0	0	0
3	2/35	2/36	1/36	0	0	0
4	2/36	2/36	2/36	1/36	0	0
5	2/36	2/36	2/36	2/36	1/36	0
6	2/36	2/36	2/36	2/36	2/36	1/36

Correlation coefficient(X,Y) = $Cov(X , Υ)] Var(X)Var(Y))$

= $E(XY - E(X) E(Y) (Var(X)Var(Y))$

p.m.f of X is

X	1	2	3	4	5	6
P(X)	11/36	9/36	7/36	5/36	3/36	1/36

p.m.f of Y is

Y	1	2	3	4	5	6
P(Y)	1/36	3/36	5/36	7/36	9/36	11/36

Using the joint probability mass function E(XY) = $XY x P(XY)$

=441/36 = 12.25

Using the probability mass functions of X and Y we get

E(X) = $Xх Р(X)$

=91/36

E(Y)= $Y x P(Y)$

=161/36

Var(X) = E(X²)-[E(X)]²

⁼ 301/36-[91/36]²

= 1.971451

Var(Y) = E(Y²)-[E(Y)]²

=764/36-[161/36]²

=1.221451

Covariance(XY) =COV(XY)

=E(XY)-E(X)E(Y)

=441/36-(91/36)X(161/36)

=0.945216

Correlation coefficient(XY) = 0.945216 V1.971451 x 1.221451

= 0.609116

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