So, there is a remark about isolated vertices distribution in my texts. I am curious if it can be proven easily or not?
How to show that if $p(n) = \frac{\ln n}n + \frac{α} n$ where $α$ is a constant, the number of isolated vertices has asymptotically a Poisson distribution with mean $λ = e^{−α}$. i.e., for each fixed $k$, $Pr[X = k] → \frac{e^{−λ}λ^k}{ k!}$ as $n → ∞$, where $X$ is the number of isolated vertices. .
It's not too hard if you approve a well-know theorem without proof.
Hint:
There is a well-known theorem that states if moments of a random variables tend to moments of a Poisson distribution then the R.V. itself is tending too.
So, all you have to do is calculating $\mathbb{E}X^i$ , but for technical reasons it's better to prove for $\mathbb{E}[X(X-1)...(X-i+1)]$