Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$. Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?
Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?