Poisson distribution with conditional probability

In summary, the conversation discusses computing conditional probabilities of a Poisson distribution, specifically P(X > x1 ∩ X > x2) with x1 > x2. It is mentioned that P(X > x1 ∩ X > x2) = P(X > x1), but this may not be true due to the memorylessness property of exponential distributions. The conversation also touches on the independence of events and how Bayes theorem can be used to find the probability of X > 80 given X > 70. Additionally, the claim is made that the only memoryless discrete probability distributions are geometric distributions.
  • #1
Woolyabyss
143
1
Hi guys,
I have a question about computing conditional probabilities of a Poisson distribution.
Say we have a Poisson distribution P(X = x) = e^(−λ)(λx)/(x!) where X is some event.
My question is how would we compute P(X > x1 | X > x2), or more specifically P(X> x1 ∩ X > x2) with x1 > x2?
I originally thought that P(X > x1 ∩ X > x2) = P(X > x1) but recently read about the memorylessness property of exponential distributions and I'm not sure if it applies to Poisson distributions.
 
Physics news on Phys.org
  • #2
Woolyabyss said:
I originally thought that P(X > x1 ∩ X > x2) = P(X > x1)

If x1 > x2 then ##\{X: X > x1, X > x2\}## is the same event as ##\{X:X>x1\}## , isn't it?
 
  • #3
Stephen Tashi said:
If x1 > x2 then ##\{X: X > x1, X > x2\}## is the same event as ##\{X:X>x1\}## , isn't it?
Yes. Also I know that the events X = x of a Poisson distribution are independent of one another but surely P(X >= 70) and P(X >= 80) for example can't be, because given at least 70 events happen, the probability that at least 80 events happen would be 10 no?
 
  • #4
Woolyabyss said:
but surely P(X >= 70) and P(X >= 80) for example can't be, because given at least 70 events happen, the probability that at least 80 events happen would be 10 no?

But my remark wasn't about the independence of events. If ##A \subset B ## then ##Pr(A \cap B) = Pr(A)##.

As far as independence goes, in most cases if ##A \subset B## then ##A## and ##B## are not independent events. Exceptions would be cases like ##Pr(A) = Pr(B) = 0 ## or ##Pr(A) = Pr(B) = 1 ##.

To find ##Pr(X > 80 | X > 70)##, what does Bayes theorem tell you ?

In the current Wikipedia article on "Memorylessness" https://en.wikipedia.org/wiki/Memorylessness there is the interesting claim:

The only memoryless discrete probability distributions are the geometric distributions, which feature the number of independent Bernoulli trials needed to get one "success," with a fixed probability p of "success" on each trial. In other words those are the distributions of waiting time in a Bernoulli process.

- surprising (to me), if true.
 
  • Like
Likes Woolyabyss

What is the Poisson distribution with conditional probability?

The Poisson distribution with conditional probability is a mathematical concept used to model the probability of a certain number of events occurring within a fixed interval of time or space, given that another event has already occurred. It is often used in statistical analysis and can provide valuable insights into the likelihood of rare events.

How is the Poisson distribution with conditional probability calculated?

The Poisson distribution with conditional probability is calculated using the formula P(X=x|Y=y) = (λ^(x+y) * e^(-λ)) / (x! * y!), where λ is the average rate of occurrence and x and y are the number of events occurring in the given interval of time or space. This formula takes into account the conditional probability of the second event occurring, given that the first event has already occurred.

What is the relationship between the Poisson distribution and conditional probability?

The Poisson distribution and conditional probability are closely related, as the Poisson distribution takes into account the conditional probability of an event occurring given that another event has already occurred. In other words, the Poisson distribution is a way to model the probability of rare events, taking into account the effects of previous events.

When is the Poisson distribution with conditional probability used?

The Poisson distribution with conditional probability is often used in situations where the occurrence of rare events is of interest, and the probability of these events may be influenced by previous events. It is commonly used in fields such as finance, insurance, and biology, where the occurrence of rare events can have significant impacts.

What are some limitations of the Poisson distribution with conditional probability?

While the Poisson distribution with conditional probability can be a useful tool for modeling rare events and their probabilities, it does have some limitations. It assumes that the events being studied are independent and that the rate of occurrence remains constant over time, which may not always be the case in real-world situations. Additionally, it may not be an accurate representation of events with extremely high or low probabilities.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
14
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
12
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
906
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
Back
Top