Suppose $N_t$ = # events in the interval $[0,T]$. The time for each event to occur follows an exponential distribution with parameter $\lambda$ i.e. $E_i \sim \exp(\lambda)$.
\begin{align} P(N_t =k) = P(\sum_{i=1}^{k}E_i\leq t, \sum_{i=1}^{k+1}E_i> t). \end{align} If each $E_i$ is continuous random variable, we have that \begin{align} P(N_t =0) &= P(E_1> t)=e^{-\lambda t}\\ P(N_t =1) &= P(E_1\leq t, E_1+E_2> t)=P(E_1\leq t)P(E_1+E_2> t)\\ P(N_t =2) &= P(E_1+E_2\leq t, E_1+E_2+E_3> t)=P(E_1+E_2\leq t)P(E_1+E_2+E_3> t). \end{align} I am kind of confused on how to compute the other probabilities i.e. from $P(N_t=1)$. Please I would appreciate an idea on how to do this