Mean over variance of a nested uniform

Jimmy picks $x \sim \text{Unif}(0,1)$ . Then Jon picks $Y \sim \text{Unif}(x, 1)$ . Compute $\dfrac{\mathbb{E}[Y]}{\operatorname{Var}(Y)}$ .

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$\mathbb{E}[Y\mid x]=\tfrac{1+x}{2}\Rightarrow \mathbb{E}[Y]=\tfrac34$ . Use total variance for $\operatorname{Var}(Y)$ .
$\mathbb{E}[Y^2]=\tfrac{11}{18}$ , so $\operatorname{Var}=\tfrac{7}{144}$ and the ratio is $\tfrac{108}{7}$ .

Answer

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15.4286 (± 0.02)

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