Question

Each student arrives in office hours one by one, independently of each other, at a steady rate. On average, three students come to a two-hour office hour time block. Let S be the number of students to arrive in a two-hour office hour time block.

What is the distribution of S? What is its parameter?

Group of answer choices

S ∼ Geo(1/3)

S∼Pois(3)

S∼Bin(2,1/3)

S-Pois(1.5)

What is the probability that S = 4?

What is the probability that S ≤ 2?

What is the variance of S?

What is the expected value of S?

Answer 1

We are given, S is the number of students to arrive in a two-hour office hour time block.

Since, the number of students to arrive in the two-hour office hour time block has the following properties:

a) can range from 0 to

b) takes integer values only

c) on average takes the value 3

Then we should have S ∼ Pois(3) [as for Poisson distribution parameter = mean]

Answer: The distribution of S is Poisson (3).

Since, S ∼ Pois(3), the probability mass function of S is given by,

Therefore, the probability that S = 4

= 0.168 [by using scientific calculator]

The probability that S ≤ 2

= 0.0498 + 0.1494 + 0.224 [by using scientific calculator]

= 0.4232

We know, for Poisson distribution E (S) = Var (S) = parameter of the distribution

Therefore, the variance of S = Var (S) = 3

And the expected value of S = E (S) = 3

Answer: The probability that S = 4 is 0.168. The probability that S ≤ 2 is 0.4232. Both the variance and expectation of S have the value 3.