Question

1. In ice hockey, play progresses until the referee blows the whistle, which could happen at any moment in the game. Whistles are an important part of the game because they offer time for teams to switch players, for broadcasters to air commercials, for fans to find or leave their seats, etc. As such, as the new NHL statistics intern, you have been tasked with studying issues related to whistles. Suppose that 8.3 whistles happen per period in hockey. (A hockey period is 20 minutes.)

c. You’re a goalie who uses whistle breaks to take a drink of water. It’s best if these occur between 3 and 5 minutes since the last whistle (not too soon, not too long!). What’s the probability the next whistle will meet your needs? How many future whistles will you have to wait, on average, to get a whistle wait time that meets your needs? (Again, you can pretend a period goes on forever for this part.)

d. Your friend collected a data set that contains the times (measured in minutes into the first period, NOT the time since the last whistle) at which random whistles were blown. Argue why it is reasonable to model these data using a uniform distribution, and give the particular pdf that would be appropriate for this setting.

Answer 1

Let X be the Random Variable denoting the time till next whistle.

So X~exp(a). Now 8.3 whistles happen in a 20 minute period.So 1 whistle happens in a 20/8.3=2.41 minutes.

So a=2.41. here a=E(X)

We need to find P(3<X<5)=exp(-3/a) - exp(-5/a)=0.16.

Part 2:

So one whistle meets this criteria with probability 0.16. So 1/0.16=6.25~7 whistles are needed to actually meet this criteria.

Part 3:

Let Y be the Random Variable denoting times at which whistles are blown. Now the referee may blow the whistle at any moment. So Y can take any value between 0 and 20 minutes depending on when the referee blows the whistles. So Suppose the referee decides to play 5 whistles. He will draw 5 numbers from U(0,20) and blow whistles at those points. Thus the distribution is U(0,20).