finding efficiently the correlation cofactor in probability

Question

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 $x_2$

my attempt:

so according to the details, i think that in order to find the correlation cofactor, i need to solve: $p(x_1,x_2)=\frac{cov(x_1,x_2)}{\sqrt{var(x_1)var(x_2)}}$, so i get that $p(x_1,x_2)=\frac{E[x_1x_2]-E[x_1]E[x_2]}{\sqrt{((E[x_1^2]-(E[x_1])^2)(E[x_2^2]-(E[x_2])^2)}}$, and it becomes a huge mess. is there a smart or efficient way to solve it using the given data instead of a lot of calculations?

would really appreciate your help and insights.

thank you very much!

Answer 1

Covariance is bilinear and symmetric.

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If $X,Y$ are random variables with $X=aY+b$ where $a,b$ are constants then: $$\mathsf{Var}X=\mathsf{Var}(aY+b)=a^2\mathsf{Var}Y$$

and:$$\mathsf{Cov}(X,Y)=\mathsf{Cov}(aY+b,Y)=a\mathsf{Cov}(Y,Y)=a\mathsf{Var}Y$$

If $a\neq0$ and $Y$ is not degenerated then this leads to:$$\rho(X,Y)=\frac{\mathsf{Cov}(X,Y)}{\sqrt{\mathsf{Var}X}\sqrt{\mathsf{Var}Y}}=\frac{a\mathsf{Var}Y}{\sqrt{a^2\mathsf{Var}Y}\sqrt{\mathsf{Var}Y}}=\frac{a}{|a|}$$

Apply this on $X_1=(-1)X_2+21$.

Answer 2

Before we answer your question: consider the random variable $X$ that follows $Bin(n,p)$ . What we get is that $E(X) = np$ and that $V(X) = np(1-p)$. In this particular case,$X_1$ follows $Bin(21,1/3)$ and $X_2$ follows $Bin(21,2/3)$. You can easily calculate now $E(X_1),V(X_1),E(X_2),V(X_2)$.All you have to find now is $E(X_1X_2)$,but in this particular case you can notice that $X_2 = 21- X_1$ and replace it above.