We observe a Poisson process with unknown rate. The rate lambda 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 lambda= 2
0.4407
To determine the conditional probability that lambda = 2 given the observed 9 arrivals over the time interval 0 to 3, we can use Bayes' theorem.
Bayes' theorem states that the conditional probability of an event A given an event B can be calculated as:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, event A represents lambda = 2, and event B represents the observed 9 arrivals over the time interval 0 to 3.
To calculate P(B|A), we need to consider the probability of observing 9 arrivals given that lambda = 2. This can be calculated using the Poisson probability formula:
P(B|A) = (e^(-lambda) * lambda^k) / k!
where lambda is the rate and k is the observed number of arrivals.
Given lambda = 2 and k = 9, we can calculate P(B|A) as follows:
P(B|A) = (e^(-2) * 2^9) / 9!
Next, we need to calculate P(A), which is the prior probability of lambda = 2. Since lambda can be either 2 or 4 with equal probability, P(A) = 0.5.
Finally, we need to calculate P(B), which is the probability of observing 9 arrivals over the time interval 0 to 3. We can calculate this by considering both possible values of lambda and summing up their respective probabilities.
P(B) = P(B|lambda = 2) * P(lambda = 2) + P(B|lambda = 4) * P(lambda = 4)
Using the same Poisson probability formula, we can calculate P(B|lambda = 4) with lambda = 4 and k = 9:
P(B|lambda = 4) = (e^(-4) * 4^9) / 9!
Since P(lambda = 4) is also 0.5, we can calculate P(B):
P(B) = P(B|lambda = 2) * 0.5 + P(B|lambda = 4) * 0.5
Finally, we can substitute all these values into Bayes' theorem, using the calculated values of P(B|A), P(A), and P(B), to find the conditional probability P(A|B) that lambda = 2 given the observed 9 arrivals over the time interval 0 to 3.
0.01416
0.88139
the denominator gets weighed by lamda (equal probability 1/2)