Introduction to Probability · Problem 5.1.11

Distributions and Densities — Grinstead & Snell

Mathematics — Probability · poisson-distribution

Problem

Suppose that

X

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

X

could be modelled using a Poisson distribution with parameter

\lambda

, where this parameter represents the average number of incoming calls per minute. Now suppose that

Y

is a random variable which represents the number of incoming calls in an interval of length

t

. Show that the distribution of

Y

is given by

P(Y = k) = e^{-\lambda t} \frac{(\lambda t)^k}{k!},

i.e.,

Y

is Poisson with parameter

\lambda t

. Hint: Suppose a Martian were to observe the police station. Let us also assume that the basic time interval used on Mars is exactly

t

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

Y

Related theory

Poisson Brackets And Poisson Manifolds

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