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As a (perhaps) less offensive twist on this question, suppose that $z_i$ are independent $p$-variate standard normals: $z_i \sim \mathcal{N}\left(0, I\right).$ Let $a$ be an unknown $p$-vector. Suppose you observe $b=a + z_1$. From this observed $b$, how would you compute an $\alpha$ prediction interval on the product $\left(a + z_1\right)^{\top}\left(a + z_2\right)$?

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