How can i find the correlation cofactor? (details inside)

would appreciate your help with this question:

a regular die is being thrown 21 times. we define:

#x_1# - the number of throws we obtained 1 or 2.

#x_2# - the number of throws we obtained 3,4,5,6.

how can i calculate the correlation cofactor between #x_1# and #x_2#?

trying to see if there's a smart way to calculate it rather than #p(x_1,x_2)=\frac{E[x_1x_2]-E[x_1]E[x_2]}{\sqrt{((E[x_1^2]-(E[x_1])^2)(E[x_2^2]-(E[x_2])^2)}}#?

1 Answer
MattyMatty
Jun 20, 2018

#"There is a shortcut indeed"#
#"The correlation cofactor is -1."#

Explanation:

#"Obviously, we must use the fact that"#
#x_1 + x_2 = 21#
#=> x_2 = 21 - x_1#
#=> E[x_2] = 21 - E[x_1]#

#=> E[x_1 x_2] = E[x_1 (21 - x_1)] = 21 E[x_1] - E[x_1^2]#
#=> E[x_1] E[x_2] = 21 E[x_1] - (E[x_1])^2#
#"So the numerator of the correlation factor becomes"#
#(E[x_1])^2 - E[x_1^2] = -var[x_1]#

#"For the denominator we have"#
#E[x_2^2]= E[(21 - x_1)^2] = 21^2 - 42 E[x_1] + E[x_1^2]#
#(E[x_2])^2 = (21 - E[x_1])^2 = 21^2 - 42 E[x_1] + (E[x_1])^2#
#"So the denominator becomes"#
#sqrt((E[x_1^2] - (E[x_1])^2)^2) = E[x_1^2] - E[x_1]^2 = var[x_1]#

#=> " correlation cofactor " = - (var[x_1]) / (var[x_1]) = -1#

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