Covariance between a chi-square and a scaled normal

Question

Let $Z\sim N(0,1)$ be independent of $Y\sim\chi^2_\nu$ , and set $W=Z/\sqrt{Y}$ . Find $\operatorname{Cov}(Y,W)$ .

Short answer

$0$ . By independence, $\mathbb{E}[YW]=\mathbb{E}[\sqrt Y]\,\mathbb{E}[Z]=0$ and $\mathbb{E}[W]=\mathbb{E}[Y^{-1/2}]\,\mathbb{E}[Z]=0$ (finite for $\nu>1$ ), so the covariance vanishes, even though $Y$ and $W$ are not independent.

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Lessons behind this question

Covariance and Correlation
Independence
The Normal Distribution

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