I can't understand the following question:
"The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} are the ordered arrival times of a homogeneous Poisson process with rate $\lambda = \frac{0.8}{T}$.
Find the probability $P[x(t) \le 1]$."
I can't understand the question. The process $x(t)$ should be a filtered Poisson process, but how can I say that the $\mathbf sum$ of the random rect signals is equal to a certain value??
Thanks in advance, any help will be appreciated.
Each $x(t)$ depends on the random sequence $(\tau_n)$ hence $x(t)$ is random. For example, the event $[x(t)=0]$ corresponds to every rectangular function in the series evaluating to $0$, that is, to the event that $|t-\tau_n|\gt\frac12T$ for every $n$. Considering the random set $\mathfrak T=\{\tau_n\mid n\in\mathbb Z\}$, one gets $$[x(t)=0]=[\mathfrak T\cap[t-\tfrac12T,t+\tfrac12T]=\varnothing], $$ from which $P(x(t)=0)$ can be deduced since the size of the random set $\mathfrak T\cap[t-\tfrac12T,t+\tfrac12T]$ is a Poisson random variable with known parameter. Can you do the same for the event $[x(t)\leqslant1]$?