  The Erd\H{o}s-R\'enyi random graph is the simplest model for node degree
distribution, and it is one of the most widely studied. In this model, pairs of
$n$ vertices are selected and connected uniformly at random with probability
$p$, consequently, the degrees for a given vertex follow the binomial
distribution. If the number of vertices is large, the binomial can be
approximated by Normal using the Central Limit Theorem, which is often allowed
when $\min (np, n(1-p)) > 5$. This is true for every node independently.
However, due to the fact that the degrees of nodes in a graph are not
independent, we aim in this paper to test whether the degrees of per node
collectively in the Erd\H{o}s-R\'enyi graph have a multivariate normal
distribution MVN. A chi square goodness of fit test for the hypothesis that
binomial is a distribution for the whole set of nodes is rejected because of
the dependence between degrees. Before testing MVN we show that the covariance
and correlation between the degrees of any pair of nodes in the graph are
$p(1-p)$ and $1/(n-1)$, respectively. We test MVN considering two assumptions:
independent and dependent degrees, and we obtain our results based on the
percentages of rejected statistics of chi square, the $p$-values of Anderson
Darling test, and a CDF comparison. We always achieve a good fit of
multivariate normal distribution with large values of $n$ and $p$, and very
poor fit when $n$ or $p$ are very small. The approximation seems valid when $np
\geq 10$. We also compare the maximum likelihood estimate of $p$ in MVN
distribution where we assume independence and dependence. The estimators are
assessed using bias, variance and mean square error.
