Poisson probability distribution

[Maybe_Memorie]

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?

 

[Ray Vickson]

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

 

[Maybe_Memorie]

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.

 

[Ray Vickson]

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

 

[Maybe_Memorie]

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

RGV

I still have no idea how to calculate this.

 

[Ray Vickson]

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.

 

[HallsofIvy]

The "inter-detection time" is the time between two consecutive clicks of the counter.

 

[Maybe_Memorie]

It's "Introduction to Probability and Statistics for Engineers and Scientists".

And actually, I have read the book.

 

[Maybe_Memorie]

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.

 

[I like Serena]

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!

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

 

[Maybe_Memorie]

Hi Maybe_Memorie!

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

Hi!

The inter-detection time is given by the exponential distribution with mean 1/lamda, yes?

 

[I like Serena]

Yes!

 

[Maybe_Memorie]

Thanks!

 

[I like Serena]

Do you have any other hanging threads?