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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?

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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 $\frac{2(n-1)}{n}p(1-p)$.

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