Poisson Processes and Bayes' Rule
We observe a Poisson process with unknown rate. The rate 饾渾 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 饾渾=2 .
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To determine the conditional probability that 饾渾=2, given that we observed exactly 9 arrivals in the time interval [0,3] and the rate 饾渾 of the Poisson process is either 2 or 4, with equal probability, we can use Bayes' rule.
Bayes' rule states that the conditional probability of an event A given event B can be calculated as:
饾憙(饾惔|饾惖) = 饾憙(饾惖|饾惔) * 饾憙(饾惔) / 饾憙(饾惖)
In this case, our event A is 饾渾=2 (the rate of the Poisson process is 2) and event B is observing exactly 9 arrivals in the time interval [0,3].
Let's break down the components of Bayes' rule:
饾憙(饾惖|饾惔): This is the probability of observing exactly 9 arrivals given that 饾渾=2. To calculate this probability, we can use the Poisson distribution with a rate of 2. The probability mass function (PMF) of the Poisson distribution is given by:
饾憙(饾憢 = 饾憳) = 饾渾^饾憳 * 饾憭^(-饾渾) / 饾憳!
where 饾憢 is the random variable representing the number of arrivals in a given interval, 饾渾 is the rate of the Poisson process, and 饾憳 is the observed number of arrivals. Plugging in the values, we get:
饾憙(9 arrivals|饾渾=2) = (2^9 * 饾憭^(-2)) / 9!
饾憙(饾惔): This is the prior probability of the rate 饾渾 being 2, which is equal to 0.5 since both rates (2 and 4) have equal probability.
饾憙(饾惖): This is the probability of observing exactly 9 arrivals, regardless of the rate 饾渾. We can calculate this by considering both possible values of 饾渾 (2 and 4) and summing up the individual probabilities:
饾憙(9 arrivals) = 饾憙(9 arrivals|饾渾=2) * 饾憙(饾惔) + 饾憙(9 arrivals|饾渾=4) * 饾憙(饾惔')
where 饾憙(饾惔') is the probability of the rate 饾渾 being 4, which is also equal to 0.5.
Now, we can substitute all these values into Bayes' rule to calculate the conditional probability 饾憙(饾惔|饾惖) = 饾憙(饾渾=2|9 arrivals):
饾憙(饾渾=2|9 arrivals) = 饾憙(9 arrivals|饾渾=2) * 饾憙(饾惔) / 饾憙(9 arrivals)
Substituting the values, we get:
饾憙(饾渾=2|9 arrivals) = (饾憙(9 arrivals|饾渾=2) * 饾憙(饾惔)) / 饾憙(9 arrivals)
Simplifying and plugging in the previously calculated probabilities, we get:
饾憙(饾渾=2|9 arrivals) = ( (2^9 * 饾憭^(-2)) / 9! ) * 0.5 / 饾憙(9 arrivals)
To calculate the final value, we need to calculate 饾憙(9 arrivals). By summing up the probabilities of observing exactly 9 arrivals for both possible rates 饾渾=2 and 饾渾=4, we get:
饾憙(9 arrivals) = 饾憙(9 arrivals|饾渾=2) * 饾憙(饾惔) + 饾憙(9 arrivals|饾渾=4) * 饾憙(饾惔')
Calculating the individual probabilities 饾憙(9 arrivals|饾渾=2) and 饾憙(9 arrivals|饾渾=4) using the Poisson distribution and plugging in the respective values, we can calculate 饾憙(9 arrivals). Then, we can substitute this value into the previous equation to calculate 饾憙(饾渾=2|9 arrivals).
This is the step-by-step process to determine the conditional probability that 饾渾=2, given the observed information and using Bayes' rule.