Mean and variance of linear combination

A question is this type if and only if it asks for the mean and/or variance of a linear combination of independent Poisson variables (e.g. C = aX + bY), and may ask whether the result itself follows a Poisson distribution.

1 questions · Challenging +1.2

5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.02m Poisson: mean = variance = lambda5.04a Linear combinations: E(aX+bY), Var(aX+bY)
Sort by: Default | Easiest first | Hardest first
OCR S3 2015 June Q5
11 marks Challenging +1.2
5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.
  1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
  2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
  3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.