Poisson Probability Distribution Problem

[Callix]

Homework Statement

An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.4 year.

a). How many loads can be expected to occur during a 4-year period?

b). What is the probability that more than 13 loads occur during a 4-year period?

c). How long must a time period be so that the probability of no loads occurring during that period is at most 0.3?

Homework Equations

The Attempt at a Solution

I tried setting up the equation as a Poisson probability distribution for a). as (e^(-0.4)*0.4^(4)) / (4!) but I wasn't sure if this was correct. I couldn't move on to b or c without knowing a. If anyone could help give me some direction with good details that would be appreciated! I want to be able to learn the material and reasoning, not simply obtain the answer.

Also, I apologize if this is not in the right category. I didn't see any homework-related sub-forums for probability and stats.

Thanks in advance!

 

[Callix]

Is a). actually just 4/0.4?

 

[Ray Vickson]

Homework Statement

An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.4 year.

a). How many loads can be expected to occur during a 4-year period?

b). What is the probability that more than 13 loads occur during a 4-year period?

c). How long must a time period be so that the probability of no loads occurring during that period is at most 0.3?

Homework Equations

The Attempt at a Solution

I tried setting up the equation as a Poisson probability distribution for a). as (e^(-0.4)*0.4^(4)) / (4!) but I wasn't sure if this was correct. I couldn't move on to b or c without knowing a. If anyone could help give me some direction with good details that would be appreciated! I want to be able to learn the material and reasoning, not simply obtain the answer.

Also, I apologize if this is not in the right category. I didn't see any homework-related sub-forums for probability and stats.

Thanks in advance!

Your $\tau = 0.4$ yr. is a time, not a rate. The parameter in the Poisson distribution is dimensionless:
$$\text{parameter} = \text{mean number} = \text{rate} \times \text{time}.$$
In your problem, the rate is $\lambda = 1/\tau = 1/0.4 = 2.5$ events per year.

Problems in probability and/or statistics are usually posted here or in the "Calculus and Beyond" forum, depending on the level of the question and the mathematical tools needed to deal with it. Occasionally they appear in the Elementary or Advanced Physics forums, especially if they have something to do with experimental error analysis or statistical mechanics and the like.

 

[Callix]

Your $\tau = 0.4$ yr. is a time, not a rate. The parameter in the Poisson distribution is dimensionless:
$$\text{parameter} = \text{mean number} = \text{rate} \times \text{time}.$$
In your problem, the rate is $\lambda = 1/\tau = 1/0.4 = 2.5$ events per year.

Problems in probability and/or statistics are usually posted here or in the "Calculus and Beyond" forum, depending on the level of the question and the mathematical tools needed to deal with it. Occasionally they appear in the Elementary or Advanced Physics forums, especially if they have something to do with experimental error analysis or statistical mechanics and the like.

Ah, that makes sense, so in this case what is x? Or is x the value that I am solving for?

 

[Callix]

So far my equation is $\frac{e^{-10}10^x}{x!}$

 

[Callix]

Oh, so $\mu$ (aka $2.5 \times 4$) is the expected value, so that should be the value that I'm looking for for a)?

 

[Ray Vickson]

Ah, that makes sense, so in this case what is x? Or is x the value that I am solving for?

There was no letter "x" in the above; there was the multiplication sign $\times$. Sorry if that confused you.

 

[Callix]

There was no letter "x" in the above; there was the multiplication sign $\times$. Sorry if that confused you.

I was referring to the equation that I made in the following post. But I realized that the value that I'm looking for is $\mu$ anyway, which if I'm understanding correctly, should be 10.

I'm confused about how to go about b). It gave me a table of CDF values listed for $\mu$ and x, so and it wants P(X>13). My logic is that that would simply sum to infinity with infinite terms, so I did 1-P(X≤13). Since the table is CDF, that would mean I simple take 1-P(13) wouldn't it?

 

[Ray Vickson]

I was referring to the equation that I made in the following post. But I realized that the value that I'm looking for is $\mu$ anyway, which if I'm understanding correctly, should be 10.

I'm confused about how to go about b). It gave me a table of CDF values listed for $\mu$ and x, so and it wants P(X>13). My logic is that that would simply sum to infinity with infinite terms, so I did 1-P(X≤13). Since the table is CDF, that would mean I simple take 1-P(13) wouldn't it?

Yes, that would be the way to do it.

 

[Callix]

Yes, that would be the way to do it.

Alright. And for C, do I just solve for $\mu$ from the Poisson equation? And then divide by the rate?

 

[Callix]

Hmm, or could I solve C using the CDF tables?

 

[Ray Vickson]

Alright. And for C, do I just solve for $\mu$? And then divide by the rate?

Solve for $\mu$ how? What would be the equation you want to solve?

 

[Callix]

Well it tells me that the probability would be 0.3 so I was thinking I could set it up as
$\frac{e^{-\mu}\mu^x}{x!}=0.3$
But then again I don't have a specified x...

 

[Callix]

Oh wait x would just be 0 because it's referring to no loads

 

[Ray Vickson]

Oh wait x would just be 0 because it's referring to no loads

Yes, exactly.

 

[WWGD]

I know this is a necropost, but just curious as to how the constant $\lambda$ is determined in real life.

 

[Mark44]

I know this is a necropost, but just curious as to how the constant $\lambda$ is determined in real life.

It's the expected number of events in whatever the time interval happens to be.

 

[WWGD]

It's the expected number of events in whatever the time interval happens to be.

And you use long-term data to compute it, e.g., the number of patients arriving at the emergency room?

 

[Mark44]

And you use long-term data to compute it, e.g., the number of patients arriving at the emergency room?

I would think so -- the average number of patients arriving per day or month, or whatever.

 

[WWGD]

I would think so -- the average number of patients arriving per day or month, or whatever.

Yes, thanks, I meant the average.