We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $t \in \mathbf{R}_+$. We assume that the number of impulses after time $t$ is given by a random variable $Y_t$ following a Poisson distribution with parameter $t$. For all $j \in \llbracket 1;N \rrbracket$ we denote by $A_{t,j}$ the event ``the system is in state $j$ after time $t$''. Justify that $P(A_{t,j}) = H_t[1,j]$.
We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$.\\
Let $t \in \mathbf{R}_+$. We assume that the number of impulses after time $t$ is given by a random variable $Y_t$ following a Poisson distribution with parameter $t$. For all $j \in \llbracket 1;N \rrbracket$ we denote by $A_{t,j}$ the event ``the system is in state $j$ after time $t$''. Justify that $P(A_{t,j}) = H_t[1,j]$.