# Expectation of degree in Bernoulli graphs

Let $$\mathcal{G}(n,p)$$ be the Bernoulli graph ditribution of $$n$$ verteces with edge probability $$p$$. It is known that the degree distribution of such graphs is the binomial distribution.

My question is the following: Let $$G$$ be a graph drawn from $$\mathcal{G}(n,p)$$ and let $$m(G)$$ be the mean degree of the graph. Then $$m(\cdot)$$ is a random variable on $$\mathcal{G}(n,p)$$. Is it known what is its distribution?

It's $$\frac{2}{n}$$ times the binomial distribution with probability $$p$$ on $$\binom{n}{2}$$ trials. Each possible edge appears with probability $$p$$. If it does, it contributes $$2$$ to the sum of degrees ($$1$$ for each endpoint) or $$\frac2n$$ to the average degree.
Mean $$(n-1)p$$, variance $$\frac{2(n-1)}{n}p(1-p)$$.