15.3 Exercise 34

Noise pulses arrrive on a data phone line according to an arrival process such that for each $t > 0$ the number N_t of arrivals in time interval $\left(0 , t\right]$, in hours, is Poisson $\left(7 t\right)$. The ith pulse has an “intensity” Y_i such that the class $\left\{Y_{i} : 1 \leq i\right\}$ is iid, with the common distribution function $F_{Y} \left(u\right) = 1 - e^{- 2 u^{2}}$ for $u \geq 0$. Determine the probability that in an eight-hour day the intensity will not exceed two.