Normal bundle of fibers

Question

Everything is defined over the field of complex numbers.

Let $Y \subset \mathbb P^{2n+1}$ be a smooth projective variety of dimension $n$. Denote by $E$ the (naive) projectivization of the normal bundle $N_{Y|\mathbb P^{2n+1}}$: $$ \pi:E=\mathbb P\left( N_{Y|\mathbb P^{2n+1}}\right) \longrightarrow Y $$ and let $F:=\pi^{-1}(y)=\mathbb P^n$ be a fiber.

Question 1. Can we describe explicitly the normal bundle of $F$ inside $E$? Namely, can we say something about $N_{F|E}$?

Question 2. Can we say something more if we know that $Y \subset \mathbb P^{2n+1}$ is a Legendrian subvariety?