[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that [math]X[/math] is a random variable which represents the

number of calls coming in to a police station in a one-minute interval. In the text, we showed that [math]X[/math] could be modelled using a Poisson distribution with parameter [math]\lambda[/math], where this parameter represents the average number of incoming calls per minute. Now suppose that [math]Y[/math] is a random variable which represents the number of incoming calls in an interval of length [math]t[/math]. Show that the distribution of [math]Y[/math] is given by

[[math]] P(Y = k) = e^{-\lambda t}{{(\lambda t)^k}\over{k!}}\ , [[/math]]

i.e., [math]Y[/math] is Poisson with

parameter [math]\lambda t[/math]. Hint: Suppose a Martian were to observe the police station. Let us also assume that the basic time interval used on Mars is exactly [math]t[/math] Earth minutes. Finally, we will assume that the Martian understands the derivation of the Poisson distribution in the text. What would she write down for the distribution of [math]Y[/math]?