Poisson probability distribution

In summary: Hi! :smile:Yes, I have another hanging thread. Would you like me to take a look?Yes, I have another hanging thread. Would you like me to take a look?
  • #1
Maybe_Memorie
353
0

Homework Statement



Homework Statement



A particle detector is set up to detect type A particles. These are detected as a poisson process with parameter lamda = 0.5 per day.

(i) What is the probability that 3 or more will be detected in anyone day?

(ii) What is the distribution of inter-detection times for these particles?

(iii) What is the probability that the inter-detection time for two consecutive particles will be less than 3 days?

(iv) The detector also detects type B particles. These occur with rate lamda = 0.5 when no type A are detected, and rate lamda = 1 when 1 or more type A are detected. If on anyone day one or more type B are detected, what is the probability one or more type A are also detected?


The Attempt at a Solution



(i) was easy. I got 0.01439

For (ii) I have absolutely no idea what to do.


For (iv), I used P(at least 1 A | at least 1 B) = (at least 1 A and 1 B)/P(at least 1 B)
I ended up getting 0.2425.
Is this correct?
 
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  • #2
Maybe_Memorie said:

Homework Statement



Homework Statement



A particle detector is set up to detect type A particles. These are detected as a poisson process with parameter lamda = 0.5 per day.

(i) What is the probability that 3 or more will be detected in anyone day?

(ii) What is the distribution of inter-detection times for these particles?

(iii) What is the probability that the inter-detection time for two consecutive particles will be less than 3 days?

(iv) The detector also detects type B particles. These occur with rate lamda = 0.5 when no type A are detected, and rate lamda = 1 when 1 or more type A are detected. If on anyone day one or more type B are detected, what is the probability one or more type A are also detected?


The Attempt at a Solution



(i) was easy. I got 0.01439

For (ii) I have absolutely no idea what to do.


For (iv), I used P(at least 1 A | at least 1 B) = (at least 1 A and 1 B)/P(at least 1 B)
I ended up getting 0.2425.
Is this correct?

For part (ii): this is standard probability theory. Do you have a probability textbook or lecture notes? I would be surprised if the answer cannot be found therein. However, if that (i.e., a relevant text or notes) does not cover your situation, you can give us more information about your situation. The real problem is that (ii) is easy, but needs some elementary but lengthy preliminary work (or else just needs cookbook quoting with no understanding attached). Of course, once you have (ii), getting (iii) is easy.

RGV
 
  • #3
Ray Vickson said:
For part (ii): this is standard probability theory. Do you have a probability textbook or lecture notes? I would be surprised if the answer cannot be found therein. However, if that (i.e., a relevant text or notes) does not cover your situation, you can give us more information about your situation. The real problem is that (ii) is easy, but needs some elementary but lengthy preliminary work (or else just needs cookbook quoting with no understanding attached). Of course, once you have (ii), getting (iii) is easy.

RGV

I'm using the book by Sheldon M. Ross

The problem is I don;t know what (ii) is actually asking.
 
  • #4
Maybe_Memorie said:
I'm using the book by Sheldon M. Ross

The problem is I don;t know what (ii) is actually asking.

It is asking for the distribution of inter-arrival times in a Poisson process.

RGV
 
  • #5
Ray Vickson said:
It is asking for the distribution of inter-arrival times in a Poisson process.

RGV

I still have no idea how to calculate this.
 
  • #6
Ross has written several Probability textbooks. You don't say which one you are using, but that matters not at all: they are all excellent and all have everything you need. Here is a hint: read the book.
 
  • #7
The "inter-detection time" is the time between two consecutive clicks of the counter.
 
  • #8
It's "Introduction to Probability and Statistics for Engineers and Scientists".

And actually, I have read the book.
 
  • #9
HallsofIvy said:
The "inter-detection time" is the time between two consecutive clicks of the counter.

I understand this. It's the time between the "n"th particle and the "n+1"th particle.

But I don't know how to find this distribution.
 
  • #10
Maybe_Memorie said:
I understand this. It's the time between the "n"th particle and the "n+1"th particle.

But I don't know how to find this distribution.

Hi Maybe_Memorie! :smile:

I see this thread is still hanging.
Do you have the answer by now?
 
  • #11
I like Serena said:
Hi Maybe_Memorie! :smile:

I see this thread is still hanging.
Do you have the answer by now?

Hi! :smile:

The inter-detection time is given by the exponential distribution with mean 1/lamda, yes?
 
  • #12
Yes! :cool:
 
  • #13
Thanks! :smile:
 
  • #14
Do you have any other hanging threads?
 

Related to Poisson probability distribution

What is a Poisson probability distribution?

A Poisson probability distribution is a mathematical model used to predict the likelihood of a certain number of events occurring within a specific time period, given the average rate at which the events occur. It is often used in situations where the events are independent and occur at a constant rate.

What are the characteristics of a Poisson probability distribution?

There are several key characteristics of a Poisson probability distribution, including:

  • The events must occur independently of each other.
  • The average rate at which the events occur must be constant.
  • The probability of an event occurring in a given time interval is proportional to the length of the interval.
  • The probability of more than one event occurring in a very small interval is considered to be negligible.

How is a Poisson probability distribution calculated?

The Poisson distribution is calculated using the formula P(x) = (e * μx) / x!, where x is the number of events, μ is the average rate of events, and e is the mathematical constant approximately equal to 2.71828. This formula gives the probability of x events occurring within a given time period.

What types of real-world applications use the Poisson probability distribution?

The Poisson probability distribution is commonly used in fields such as insurance, finance, and manufacturing to predict the likelihood of rare events or accidents occurring. It is also used in epidemiology to study the spread of diseases and in physics to model radioactive decay.

What are the limitations of the Poisson probability distribution?

The Poisson distribution assumes a constant rate of events and independence between events, which may not always hold true in real-world situations. Additionally, it is not suitable for predicting events that occur at very high rates or for modeling events that are not discrete, such as time or distance.

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