Modifying frequency to eliminate 0

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In summary: The solution eliminated the probability of 0 profit in both cases because if you have a Poisson process with a rate of 0, then the probability of making a profit in 10 minutes is 0.6065307.
  • #1
soran
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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!
 
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  • #2
soran said:
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 $P(X=0)=0.6065307$.
 
  • #3
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?
 

FAQ: Modifying frequency to eliminate 0

1. How can modifying frequency eliminate 0?

Modifying frequency refers to changing the frequency of a signal or wave. By adjusting the frequency, it is possible to eliminate any zeroes that may be present in the signal. This is because changing the frequency can shift the signal to a different part of the frequency spectrum where the zeroes are not present.

2. What is the purpose of eliminating 0 in frequency modification?

Eliminating 0 in frequency modification is done to improve the quality and clarity of the signal. Zeroes can disrupt the signal and cause distortions, so by removing them through frequency modification, the signal becomes more accurate and easier to interpret.

3. How is frequency modified to eliminate 0?

Frequency can be modified using various techniques such as filtering, amplification, and mixing. These methods involve changing the amplitude, phase, and frequency of the signal to achieve the desired frequency and eliminate any zeroes in the signal.

4. What types of signals can benefit from frequency modification to eliminate 0?

Frequency modification can be applied to a wide range of signals, including audio signals, radio signals, and electronic signals. It is especially useful in communication systems, where the elimination of zeroes can improve the overall quality of the signal.

5. Are there any drawbacks to modifying frequency to eliminate 0?

While frequency modification can improve the quality of a signal by eliminating zeroes, it can also introduce new distortions or alter the original signal in unintended ways. It is essential to carefully design and test the frequency modification process to minimize any potential drawbacks.

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