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December 7th, 2017, 04:20 AM   #1
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Talking Poisson Method?

Hi!

I have been given this task-

A restaurant based in a city centre monitors the rate at which customers
are arriving and leaving. It finds during a particular lunchtime that
customers arrive at a mean rate of 4.5 in 30 minutes, while they leave at
a mean rate of 2.2 in 30 minutes. Giving your answers to 2 d.p. calculate
the probability that


a) exactly 6 customers arrive in one 30-minute period.
[3]

b) at most 3 customers leave in one 30-minute period.
[2]

c) between 2 and 5 customers inclusive leave in one 30-minute
period.
[4]

d) at most 12 customers arrive in a 1-hour period.
[2]

e) between 5 and 7 customers inclusive leave in a 1-hour period.
[4]


Can anyone advise me on how to work it out using the Poisson method? And then I think i will be able to handle the rest of it from there? I am just unsure on how to take the first step towards answering it.

Thanks for any help it's greatly appreciated!
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December 7th, 2017, 04:06 PM   #2
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just assume that arrivals and departures are distributed with a Poisson distribution of the appropriate rate.

Read up on the distribution and see if you can figure it out. Shout back if not.
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December 11th, 2017, 08:54 AM   #3
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Hi Romsek

So,

I have been reading up on the Poisson Distribution.

I have worked out that P(x;μ) = (e-4.5) (μx) / x!

so the answer to the first question is P(x=6) = 0.128

However, how would I work it out if the question asks at most three customers and between 2 and 5 customers?

As I am not sure how I would change the calculation if I am working out at most or between numbers?

Thanks for all your help!
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December 11th, 2017, 09:35 AM   #4
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Quote:
Originally Posted by DecoMate View Post
So,

I have been reading up on the Poisson Distribution.

I have worked out that P(x;μ) = (e-4.5) (μx) / x!

so the answer to the first question is P(x=6) = 0.128

However, how would I work it out if the question asks at most three customers and between 2 and 5 customers?

As I am not sure how I would change the calculation if I am working out at most or between numbers?

Thanks for all your help!
at most 3 customers is 0 through 3 customers
2 to 5 customers is 2 through 5 customers.

You can either use the CDF of the Poisson distribution, $P[k]$ and compute

$P[3]$ in the first case and $P[5]-P[1]$ in the second.

It's probably easiest though to just sum up the individual pmf values, $p[k]$, as the CDF formula is a bit arcane.

In the first case you would just have $\sum \limits_{k=0}^3~p[k]$

In the second case $\sum \limits_{k=2}^5~p[k]$
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