You manage blue whale sightings by sea coast. The whales are known to show up at a rate of 10 per month (assume 30 days per month).

You manage blue whale sightings by sea coast. The whales are known to show up at a rate of 10 per month (assume 30 days per month). They show up independently, so the arrival of one has no impact on the probability of another showing up, and there are sufficient whales that the number of whales showing up in two non-overlapping intervals of equal length are independent and have the same distribution.

Let X represent the number of whales showing up in the next month.

Let Y represent the time between successive arrivals of whales.

1. What is the probability that more than one shows up in the next month?

2. What is the probability that the time between the next two arrivals of whales is 3 days or less?

3. Your have a birthday next month. What is the probability that no whale will show up during your birthday (note that your birthday, like all others, lasts 24 hours)?

Over the next year (12 calendar months), you are interested in W, which is the number of months that have no whales arriving

4. What is the mean of W?

5. What is the standard deviation of W?

6. What is the probability that 3 months have no whales showing up? P(W=3)=?

Over the next n=63 years, you wish to approximate the distribution for the average number of months with zero whales visiting per year. (i.e., Wbar = sum Wi/n)

7. What is the mean of Wbar?

8. What is the standard deviation of Wbar?