PoissonDistribution- Demand problems

Question

I am still not too familiar with Poisson Distribution and recently I have come across a question like the following

A grocery store purchases one item for $X$ dollars and sells it for $Y$ dollars. If the item is unsold, they can not refund it. The demand for the item varies according to a $Poisson(\lambda)$ distribution. The question is what is the number of items that the store should order from the supplier....

I can't seem to wrap my head around this type of question as I cant find a way to apply the Poisson distribution.

Can anyone elaborate on this to me?

Answer 1

Hints:

Consider the following questions - you might not want to try to find closed forms:

• If the shop buys $n$ items for $X$ each, how much does it spend?

• What is the probability that shop the sells $N$ items (with $0 \le N \lt n$)? Use the Poisson distribution here

• What is the shop's revenue if it sells $N$ items (with $0 \le N \lt n)$?

• What is the probability that shop the sells at least $n$ items? Use the Poisson distribution here and you might want to leave this as a sum

• What is the shop's revenue if it sells $n$ items?

• What is the shop's expected gross revenue if it cannot sell more than $n$ items?

• What is the shop's expected net profit if it orders $n$ items?

• What would the previous answers be if you replaced $n$ by $n+1$?

• What is the change in net profit if you replaced $n$ by $n+1$?
• For which $n$ does the change in net profit first become negative?