Problem 1: Concentration in isolation

Consider a graph $G \sim \mathcal{G}_{n,p}$ with $p=D/n$ and $D > 0$ is some fixed number. Let $X$ be the number of isolated vertices in $G$ (i.e., the number of vertices that have no neighbors). Determine $\mathbb{E}[X]$ and then show that for any $\lambda > 0$,

$$\mathbb{P}\left[|X-\mathbb{E}[X]| \geq 4 \lambda \sqrt{D n}\right] \leq O(1) e^{-\lambda^2/2}$$

[Hint: Setup an “edge exposure” martingale. You will need to control the number of exposure steps.]