Chickens at Rolling Meadows Farm lay an average of 18 eggs per day. The farmer has rigged a fancy monitoring device to the nesting boxes so that he can monitor exactly when the hens lay their eggs. Assume that no 2 eggs will be laid at exactly the same time and that the eggs (and chickens) are independent from each other. The farmer wants to know how long he will have to wait (in minutes) for the next half-dozen (6) eggs to be laid so that he can bake a chocolate cake if he starts monitoring at the first rooster crow in the morning.
a. Why is this Gamma problem? What makes this a Gamma situation instead of an Exponential situation?
b. What does X represent in this scenario?
c. What are the parameters in units matching this specific question?
d. What is the expected length of time (in minutes) the farmer will have to wait for the half-dozen eggs to be laid?
e. What is the standard deviation?
f. What is the probability density function for the wait time (in minutes) for the half-dozen eggs? Write it in function form and also graph it.
g. What is the cumulative distribution function for the wait time (in minutes) for the half-dozen eggs? Write it in function form and also graph it.
h. Find the probability the farmer will have to wait for his half-dozen eggs longer than it takes him to do his morning chores and run his errands in town (6 hours after the rooster crows).
i. Find the probability that the sixth egg will be laid while he is eating dinner (between 12 and 13 hours after the rooster crows)?