In given Example, the time it takes Alison to swim a mile on any given day was assumed to be independent of the time that it takes Julie to swim a mile on that same day. Refer to the explanation of when two random variables are independent, given earlier in this section. Explain whether or not it is reasonable to assume that Alison's and Julie's times on the same day are independent in each of the following scenarios:
1. They swim at different times, and the one who swims first does not report her time to the other until after they are both done.
2. They swim side by side in two lanes of a pool.
3. They swim at different times, but the one who swims second knows the time it took the other to swim the mile that day.
Example
Alison and her sister Julie each swim a mile every day. Alison's times are normally distributed with mean = 37 minutes and standard deviation = 1 minute. Julie is faster but less consistent than Alison, and her times are normally distributed with mean = 33 minutes and standard deviation = 2 minutes. On any day, their times are independent of each other. Can Alison ever beat Julie?