Question

Michael Jordan likes to take 3-point shots. In fact, the number of 3-point shots he takes up to
time t is a Poisson process with rate 10 per hour. We assume that a game starts at 6pm.
(a) What is the expected number of 3-point shots he takes in the first 10 minutes of a game?
(b) What is the expected number of 3-point shots he takes between 6:20pm and 6:30pm?
(c) What is the probability that he takes exactly one 3-point shot between 6:20pm and 6:30pm?
(d) What is the expected time he takes 5 3-point shots?
(e) Suppose he didn't take any 3-point shot in the first 10 minutes. Given that he has not taken
any 3-point shot in the first 10 minutes, what is the expected time it takes for him to take
the first 3-point shot?
Now suppose that Michael can make any particular 3-point shot with probability 0.4
(independently of all other shots).
(f) What it the expected number of 3-point shots he makes in the first 30 minutes of a game?
(g) What is the probability that he makes exactly two 3-point shots in the first 20 minutes?
(h) What is the probability that he makes at least three 3-point shots in the first 30 minutes?
(i) What is the expected time it takes for him to successfully score his first 3-pointer?
(j) What is the probability that he attempts three 3-point shots in the first 15 minutes and only
one of the 3-point shots is successful?
(k) What is the probability that the first five 3-point shots are all successful?
Meanwhile, independently of his 3-point shots, Michael takes 2-point shots according to a
Poisson process, and the mean time between two successive 2-point shots is 3 minutes.
(l) What is the probability that he will take his first 2-point shot before his first 3-point shot?
(m) What is the expected amount of time until he takes his third shot (for either 2 or 3 points)?
Now suppose that Michael can make any particular 2-point shot with probability 0.6
(independently of all other shots).
(n) What is the mean and variance of the number of points (from either 2-point shots or 3-point
shots) he makes during the first 20 minutes of a game?

Answer 1

(a) What is the expected number of 3-point shots he takes in the first 10 minutes of a game?

Rate = 10 per hour = 10/6 per 10 minute = 5/3 per 10 minute

So, expected number of 3-point shots he takes in the first 10 minutes of a game = 5/3 = 1.67

(b) What is the expected number of 3-point shots he takes between 6:20pm and 6:30pm?

The time interval between 6:20pm and 6:30pm is 10 minutes.

So, expected number of 3-point shots he takes in the first 10 minutes of a game = 5/3 = 1.67

(c) What is the probability that he takes exactly one 3-point shot between 6:20pm and 6:30pm?

Pr[X = 1] = exp(-) ¹ /1! = 1.67 * exp(-1.67) = 0.3144

(d) What is the expected time he takes 5 3-point shots?

Time interval between successive poisson process follow exponential distribution with rate .

Total time interval for n poisson process will follow Gamma distribution with parameters n and .

= 5 /3 per 10 minute = 5/30 per minute = 1/6 per minute

Expected time he takes 5 3-point shots = n / = 5 / (1/6) = 30 minute

(e) Suppose he didn't take any 3-point shot in the first 10 minutes. Given that he has not taken
any 3-point shot in the first 10 minutes, what is the expected time it takes for him to take
the first 3-point shot?

Due to memorylessness property of exponential distribution, time it takes for him to take the first 3-point shot will follow Exp( = 1/6)

Expected time it takes for him to take the first 3-point shot = 1 / = 1 / (1/6) = 6 minute