I am presented with a randomly generated graph $G$, and am trying to 1) calculate the expected number of isolated vertices in $G$; and 2) prove an upper bound on the probability that the expected number of isolated vertices deviates from the expected value by a certain amount.
Formally $G$ is a random graph with $n$ vertices and $cn$ edges for a constant $c > 0$. $G$ is generated as follows: let $P$ be the set of all $\binom{n}{2}$ possible pairs of vertices. Pick a pair uniformly, add the edge between said pair to $G$; then delete the pair from $P$. Repeat until $cn$ edges have been added to $G$.
Now, if we define $X$ as the number of isolated vertices in $G$. How can I calculate $E[X]$ and show that:
$$P(|X-E(X)| \geq 2\lambda \sqrt{cn}) \leq 2e^{-\lambda^2/2}$$