The Poisson process is a model for events that occur in continuous time, at a constant rate > > 0 per unit time, with events occurring independently of each other. Specifically, if X(t) is the discrete random variable recording the number of events that are observed to occur in the interval [0, t),then we have that X(t) Poisson (xt), that is1 and zero otherwise. Also, the counts of events in disjoint time intervals are probabilistically independent: for example, for intervals [0,t) and [t,t + s), the numbers of events in the two intervals, X₁ and X₂ say, have the property P\left(X_{1}=x_{1} \cap X_{2}=x_{2}\right)=P\left(X_{1}=x_{1}\right) P\left(X_{2}=x_{2}\right) X_{1} \sim \operatorname{Poisson}(\lambda t) \quad X_{2} \sim \operatorname{Poisson}(\lambda With this information, answer the following questions based on the Poisson process model and its relationship with the Poisson distribution. (a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval?1 MARK O Page visits to a particular website occur according to a Poisson process with rate parameter lambda = 20 per minute. What is the expected number of visits to the website in any given one hour period?1 MARK What is the probability that the time of the first event that is observed to occur in a Poisson process with rate à per unit time, after initiation at t = 0, occurs later than time t = to, for fixed value to? Justify your answer.2 MARKS p(x)=P(X(t)=x)=e^{-\lambda t} \frac{(\lambda t)^{x}}{x !} \quad x=0,1,2, \ldots