MHB Modifying frequency to eliminate 0

[soran]

Q: ppl arrive at Poisson rate of 0.5/min. Profit X is rand as follows:

Pr(X=0) = 0.7
Pr(X=1) = 0.1
Pr(X=2) = 0.1
Pr(X=3) = 0.1

Probability of 0 profit in 10min?

The start of the solution says to modify the Poisson to eliminate the case of 0 profit.
I don't understand why this is the case or how to identify this type of problem as such.
I don't see another way to reach the solution either...

Thanks!

 

[Jameson]

The start of the solution says to modify the Poisson to eliminate the case of 0 profit.
I don't understand why this is the case or how to identify this type of problem as such.
I don't see another way to reach the solution either...

Thanks!

Hi soran,

Welcome to MHB! :)

You can scale a Poisson random variable in the following way:

If $X \sim \text{Poisson }(\lambda)$ where $\lambda$ is some rate per hour. Then $Y \sim \text{Poisson }(k \lambda)$ where $k \lambda$ is some rate per $k$ hours. Basically if you double the interval over which you are looking, you should have on average double the occurrences.

I would try thinking of this as a Poisson process with $\lambda=5$, which corresponds to 5 something per 10 minutes. If we call this process $Y$, what is the formula for $P(Y=0)$?

EDIT: I'm actually a little unsure of how $X$ corresponds to the Poisson process. Is this the whole problem? I ask because if $X \sim \text{Poisson }(\lambda)$ then $P(X=0)=0.6065307$.

 

[soran]

Thanks it's good to be here :)

To clarify, the rate at which customers arrive at a store each min is Poisson~L but the profit that is earned from these customers is a discrete distribution X whose pdf is given.

I understand that 10*L=L* is also Poisson and L* is the param for 10min intervals. What I don't understand is the next part where the solution eliminates the probability of a profit of 0 as such:

Poisson process for non-zero profits is (10*L)[1-Pr(X=0)] = 1.5
Overall probability of zero profit is then = e^-1.5

The next question asks for the probability of making a profit of 2 in 10min.
The solution again removed the probability of 0 profit. Why is it so important to remove the probability of 0 both times?