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
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