Egg laying. 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 a 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)?
Exercise 32.1:
Egg laying. 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 egg to be laid if he starts monitoring at the first rooster crow in the morning.
a. Why is this an Exponential problem?
b. What does X represent in this scenario?
c. What is the parameter in units matching this specific question?
d. What is the expected length of time (in minutes) the farmer will have to wait for the first egg to be laid?
e. What is the standard deviation?
f. What is the probability density function for the wait time (in minutes) for the first egg? Write it in function form and also graph it.
g. What is the cumulative distribution function for the wait time (in minutes) for the first egg? Write it in function form and also graph it.
h. What is the probability the farmer will have to wait longer than it takes him to make his cup of coffee (the first 10 minutes after the rooster crows)?
i. What is the probability the first egg will be laid while he is milking the cow (between 30 minutes and an hour after the rooster crows)?
j. Given that the first egg did not come while he was making his coffee, what is the probability that the first egg will come while he is milking the cow?
k. The farmer is out doing chores and can't check the fancy egg monitor. If the farmer wants to wait to go to the henhouse to feed the hens until he is 90% sure the first egg has been laid, how long should he wait?