The number of fish a fisher can catch follows a poisson distribution. During 300 consecutive 1-hour periods there were 45 periods where no fish are caught.
Approximately how many fish are caught in 300 hours and what is the median time it will take for the fisher to catch a fish?
For the number of fish caught I thought it would be as simple as 255 fish. However, this is assuming that only 1 fish is caught in each period.
For the median I did Ln(2)/-Ln(45/300) = 0.3654
Does this logic make sense? I am much less confident in the first part.
You are expected to use the fact that no fish were caught $0.15$ of the time to find the average number of fish caught per hour. This is the $\lambda$ of a Poisson distribution that has $0.15$ chance of no events. Then multiply the average number of fish per hour by $300$ to get the expected number of fish caught in total. The median time to catch a fish will be the time interval that has no fish caught $0.50$ of the time.