Several weeks ago random value $X$ (Variance - Var($X$) and Expectation - $\mathbb{E}(X)$) were introduced in terms of our probability course. A week ago we were given problems to think about, one of them is the following: for a given graph $G(n, p)$ is constructed by removing randomly and independently edges in a complete graph with $n$ vertices, every edge is left untouched with probability $p$. Let $T_n$ be a random value characterizing number of 'triangles' in $G(n, p)$. Task is to find $\mathbb{E}(T_n)$ and Var$(T_n)$.
I've found that this is related to Erdős–Rényi model, however haven't solved it in 3 day in row already. Do you have any ideas? Especially interested in Var($X$)
Hints:
Use the linearity of expectations. There are $\ {n\choose3}\ $ triangles in the original complete graph. For $\ t=1,2,\dots,{n\choose3}\ $ define an indicator function $$ I_t=\cases{0& if one of the edges of the $\ t^\text{th}\ $triangle gets deleted\\ 1& otherwise}\ . $$ Then $\ \displaystyle T_n=\sum_{t=1}^{n\choose3}I_t\ $. Can you calculate $\ \mathbb{E}\big(I_t\big)\ $?
To get $\ \text{Var}\big(T_n\big)\ $, use the formula \begin{align} \text{Var}\big(T_n\big)&=\mathbb{E}\big(T_n^2\big)~-\mathbb{E}\big(T_n\big)^2\\ &=\mathbb{E}\left(\sum_{s=1}^{n\choose3}\sum_{t=1}^{n\choose3} I_sI_t\right) -\mathbb{E}\big(T_n\big)^2\\ &=\mathbb{E}\big(T_n\big)+2 \mathbb{E}\left(\sum_{s=1}^{{n\choose3}-1} \sum_{t=s+1}^{n\choose3} I_sI_t\right)- \mathbb{E}\big(T_n\big)^2\ . \end{align} To evaluate this formula you'll need to calculate $\ \mathbb{E}\big(I_sI_t\big)\ $ for $\ 1\le s<t\le{n\choose3}\ $. If the $\ s^\text{th}\ $ and $\ t^\text{th}\ $ triangles have no edges in common, then $\ I_s\ $ and $\ I_t\ $ are independent, so $\ \mathbb{E}\big(I_sI_t\big)=$$\mathbb{E}\big(I_s\big)\mathbb{E}\big(I_t\big)\ $. To complete the calculation you will need to: