[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]

It is often assumed that the auto traffic that arrives at the

intersection during a unit time period has a Poisson distribution with expected value [math]m[/math]. Assume that the number of cars [math]X[/math] that arrive at an intersection from the north in unit time has a Poisson distribution with parameter [math]\lambda = m[/math] and the number [math]Y[/math] that arrive from the west in unit time has a Poisson distribution with parameter [math]\lambda = \bar m[/math]. If [math]X[/math] and [math]Y[/math] are independent, show that the total number [math]X + Y[/math] that arrive at the intersection in unit time has a Poisson distribution with parameter [math]\lambda = m + \bar m[/math].