# Sum of Poisson Distributions- AS Further Mathematics Stats.

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#1
Here is the Question I'm stuck on:

A petrol station has service areas on both sides of a motorway, one to serve east-bound traffic and the other for west-bound trafﬁc. The number of east— bound vehicles arriving at the station in one minute has a Poisson distribution with mean 0.9, and the number of west-bound vehicles arriving in one minute has a Poisson distribution with mean 1.6 the two distributions being independent.

(i) Find the probability that in a one-minute period
(a) no vehicles arrive
(b) more than two vehicles arrive at this petrol station

Given that in a particular one-minute period three Vehicles arrive, find
(ii) the probability that they are all from the same direction
(iii) the most likely combination of east-bound and west-bound

I've completed part i, but don't know how to solve ii and iii. The answers should be: ii. 0.309, and iii. 1 eastbound, 2 westbound. Could someone answer this with an explained solution?- I'm teaching FM S1 to myself so the more I can learn, the better.

Thankyou
0
3 years ago
#2
For part (ii) you need to use conditional probability. Let event A denote three vehicles arriving from the same direction in a one minute period and let event B denote three vehicles arriving (from any direction) in a one minute period. Then, P(A given B) = P(A n B) / P(B).

You can find P(A) from P(3 East)P(0 West) + P(0 East)P(3 West). You can find P(B) from P(3) using lambda = (0.9 + 1.6).

For part (iii) you will need to work out P(3 East)P(0 West), P(2 East)P(1 West) etc and compare.
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