Introduction to Probability · Problem 5.1.32

Distributions and Densities — Grinstead & Snell

Mathematics — Probability · poisson-distribution

Problem

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

m

. Assume that the number of cars

X

that arrive at an intersection from the north in unit time has a Poisson distribution with parameter

\lambda = m

and the number

Y

that arrive from the west in unit time has a Poisson distribution with parameter

\lambda = \bar{m}

. If

X

and

Y

are independent, show that the total number

X + Y

that arrive at the intersection in unit time has a Poisson distribution with parameter

\lambda = m + \bar{m}