The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes?

The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. The average number of fish caught in 15 minutes to be determined.

Let X the number of fish caught in 15 minutes. Therefore the sample size X takes on the values $0,1,2,3,4,5,6,7,8,9,10\dots$

Here X follows a poisson distribution with the probability mass function of poisson distribution as

$P(X=x)=\frac{e^{-\mu }{\mu }^x}{x!}$

According to the question, the average number of fish caught in an hour is eight and the interest is the number of fish caught in 15 minutes. Therefore in 15 minutes number of fish caught by $\frac{8}{60}\times 15=2$fish.

So average of Poisson distribution is $\mu =2$fish in 15 minutes.

Therefore the required average number of fish caught in 15 minutes is 2 fish.