Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by: $$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$ Show that $$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. For a graph $H = \left( S _ { H } , A _ { H } \right)$ with $S _ { H } \subset \llbracket 1 , n \rrbracket$, the random variable $X_H$ is defined by:
$$\forall G \in \Omega _ { n } \quad X _ { H } ( G ) = \begin{cases} 1 & \text { if } H \subset G \\ 0 & \text { otherwise } \end{cases}$$
Show that
$$\mathbf { E } \left( X _ { H } \right) = p _ { n } ^ { a _ { H } } .$$