Probability question; Conditional probability and poisson distribution

In summary, the problem involves a radioactive source emitting particles according to a Poisson process at a rate of λ per unit time, with each particle having a probability p of being detected by an instrument. The number of particles emitted in a given time interval is denoted by X, and the number of those particles that are detected is denoted by Y. The conditional probability of Y being equal to r given that X is equal to k can be calculated using the formula p(Y=r|X=k) = p((Y=r) ∩ (X=k)) / p(X=k), where p(X=k) is given by the Poisson distribution formula ((μ^k)/k!)*e^(−μ). The joint probability p((
  • #1
TaliskerBA
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Homework Statement



A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given time interval of length T , and Y the number of those particles that are detected. As usual, let μ = λT and q = 1 − p.
(i) What is the conditional probability p(Y = r|X = k)?


Homework Equations



I know that I want p(Y=r|X = k) = p((Y=r) ∩ (X=k)) / p(X=k)

I know from poisson distribution that p(X=k) = ((μ^k)/k!)*e^(−μ)


The Attempt at a Solution



I don't understand how I can work out what p((Y=r) ∩ (X=k)) equals but this is my attempted solution:

Since there are k particles emitted and we want to know the probability that r of them have been detected then using binomial distribution:

p((Y=r) ∩ (X=k)) = [k choose r](p^r)(q^(k-r))

I know this is wrong but I can't quite work out how to tie it all together...
 
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  • #2
TaliskerBA said:
Since there are k particles emitted and we want to know the probability that r of them have been detected then using binomial distribution:

p((Y=r) ∩ (X=k)) = [k choose r](p^r)(q^(k-r))

I know this is wrong but I can't quite work out how to tie it all together...

You wrote down a formula assuming that k particles have been emitted. Therefore this is not the joint probability, but it is still relevant to the conditional probability that you want to compute...
 

1. What is the difference between conditional probability and unconditional probability?

Conditional probability is the likelihood of an event occurring given that another event has already occurred, while unconditional probability is the likelihood of an event occurring without any conditions or prior events.

2. How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the joint occurrence of two events by the probability of the first event.

3. What is the Poisson distribution and when is it used?

The Poisson distribution is a probability distribution that describes the number of events that occur in a specific time period or space, given a known average rate of occurrence. It is typically used to model rare events or events that occur independently of each other.

4. How is the Poisson distribution different from the binomial distribution?

The Poisson distribution is used for modeling the number of events that occur in a specific time or space, while the binomial distribution is used for modeling the number of successes in a fixed number of trials. Additionally, the Poisson distribution assumes that the events occur independently of each other, while the binomial distribution does not have this assumption.

5. Can the Poisson distribution be used for continuous data?

No, the Poisson distribution is only used for discrete data, meaning that the values can only take on whole numbers. Continuous data, on the other hand, can take on any value within a range.

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