Correlation of two face-counts

You roll a fair $6$ -sided die $n$ times. Let $X$ be the number of $1$ s rolled and $Y$ the number of $5$ s. Compute $\operatorname{Corr}(X, Y)$ .

Show hints (2)+

$(X,Y)$ are multinomial counts: $\operatorname{Var}=n p(1-p)$ each, $\operatorname{Cov}=-n p_X p_Y$ .
With $p_X=p_Y=\tfrac16$ : $\operatorname{Corr}=\dfrac{-n/36}{5n/36}=-\tfrac15$ .

Answer

Reveal answer →

-0.2 (± 0.005)

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