We observe a Poisson process with unknown rate. The rate λ of the Poisson process is either 2 or 4, with equal probability, and the actual value is not known. We observe the process over the time interval [0,3] and observe exactly 9 arrivals. Given this information, determine the conditional probability that λ=2 .
To determine the conditional probability that λ=2 given the information provided, we need to use Bayes' theorem. Bayes' theorem states that the conditional probability of an event A given event B is equal to the probability of event B given event A, multiplied by the probability of event A, divided by the probability of event B.
Let's define the events:
A: λ=2 (the rate of the Poisson process is 2)
B: 9 arrivals observed over the time interval [0,3]
We need to calculate P(A|B), the conditional probability that λ=2 given 9 arrivals.
Step 1: Calculate P(B|A), the probability of observing 9 arrivals given that λ=2.
In a Poisson process with rate λ=2, the probability of observing exactly k arrivals in a given time interval t is given by the Poisson distribution formula:
P(x=k) = (e^(-λ) * λ^k) / k!
In this case, we have k=9 and λ=2.
P(B|A) = P(x=9) = (e^(-2) * 2^9) / 9!
Step 2: Calculate P(A), the prior probability that λ=2.
Since the rate of the Poisson process is either 2 or 4 with equal probability, we have P(A) = P(λ=2) = 0.5.
Step 3: Calculate P(B), the total probability of observing 9 arrivals.
To calculate this, we need to consider the probability of observing 9 arrivals given that λ=2 and the probability of observing 9 arrivals given that λ=4.
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
P(A') = 1 - P(A) = 0.5 (since the other possibility is λ=4)
P(B|A') can be calculated similarly to P(B|A) using λ=4.
Step 4: Calculate P(A|B), the conditional probability that λ=2 given 9 arrivals using Bayes' theorem.
P(A|B) = P(B|A) * P(A) / P(B)
Plugging in the values we calculated, we can find the conditional probability.
To determine the conditional probability that λ=2, given that we observed exactly 9 arrivals over the time interval [0,3], we can use Bayes' theorem.
Bayes' theorem states that the conditional probability of an event A given an event B is equal to the probability of event B given event A, multiplied by the probability of event A, divided by the probability of event B.
In this case, event A is the event that λ=2, and event B is the event that we observed exactly 9 arrivals over the time interval [0,3].
Let's break down the problem step by step:
Step 1: Determine the probability of observing exactly 9 arrivals given that λ=2.
To calculate this probability, we can use the Poisson distribution with λ=2 and x=9:
P(X=x) = (e^(-λ) * λ^x) / x!
P(X=9) = (e^(-2) * 2^9) / 9!
Step 2: Determine the probability of observing exactly 9 arrivals given that λ=4.
Using the same approach as above, we can calculate this probability using the Poisson distribution with λ=4 and x=9:
P(X=x) = (e^(-λ) * λ^x) / x!
P(X=9) = (e^(-4) * 4^9) / 9!
Step 3: Determine the probability of λ=2 and λ=4.
Since the probabilities of λ=2 and λ=4 are equal, we can assume that each probability is 1/2.
P(λ=2) = 1/2
P(λ=4) = 1/2
Step 4: Calculate the conditional probability using Bayes' theorem.
Now we can use Bayes' theorem to calculate the conditional probability:
P(λ=2 | X=9) = (P(X=9 | λ=2) * P(λ=2)) / (P(X=9 | λ=2) * P(λ=2) + P(X=9 | λ=4) * P(λ=4))
P(λ=2 | X=9) = ((e^(-2) * 2^9) / 9! * (1/2)) / ((e^(-2) * 2^9) / 9! * (1/2)) + ((e^(-4) * 4^9) / 9! * (1/2))
By calculating the numerator and denominator separately and then dividing the numerator by the denominator, we can determine the conditional probability that λ=2.