Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable defined on $\mathcal{E}_n$ such that for $G \in \Omega_n$, the integer $X_n^0(G)$ equals the number of copies of $G_0$ contained in $G$. Express $X _ { n } ^ { 0 }$ using random variables of the type $X _ { H }$, and show that : $$\mathbf { E } \left( X _ { n } ^ { 0 } \right) = \sum _ { H \in \mathcal { C } _ { 0 } } \mathbf { P } ( H \subset G ) \leq n ^ { s _ { 0 } } p _ { n } ^ { a _ { 0 } } .$$
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable defined on $\mathcal{E}_n$ such that for $G \in \Omega_n$, the integer $X_n^0(G)$ equals the number of copies of $G_0$ contained in $G$.\\
Express $X _ { n } ^ { 0 }$ using random variables of the type $X _ { H }$, and show that :
$$\mathbf { E } \left( X _ { n } ^ { 0 } \right) = \sum _ { H \in \mathcal { C } _ { 0 } } \mathbf { P } ( H \subset G ) \leq n ^ { s _ { 0 } } p _ { n } ^ { a _ { 0 } } .$$