Expected number of connections in a random graph

Question

knowing that:

"It is called pure random graph to a graph of n nodes in which between each pair of nodes there is a connection with probability p and there is no such edge with probability 1-p."

How many connections has, on average, a pure random graph of n nodes?

Thanks.

Answer 1

Let $X$ be the number of choosen edges and let $X_i$ be indicator variable for edge $i$. So we have $$X = X_1+X_2+...+X_{n\choose 2}$$ Since $E$ is linear and $$E(X_i) = 0\cdot P(X_i=0)+1\cdot P(X_i=1) =0\cdot (1-p)+1\cdot p = p$$ we have $$E(X)= {n\choose 2}\cdot p$$