# Prediction Interval on a Product

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 $$\left(a + z_1\right)^{\top}\left(a + z_2\right)$$?