Question

The day before biology homework is due students come to the study room hours at an average rate of 20 students per hour. Students arrive independently to the study room.

a) Let A be the number of students arriving at the study room between 11:30 am and 11:45 am. Find the probability that at least one student comes to the study room during this period. What are the distribution, parameter(s), and support of A?

b) Given that at least one student came between 11:30 am and 11:45 am, what is the probability that fewer than 4 students came during that period?

c) What is the probability that 3 students come between 11:30 am and 11:45 am and 8 students come between 11:45 am and 12:15 pm?

d) The study room is open for 3.5 hours on the day before the last homework was due. What is the expected number of students that came to the study room that day?

Answer 1

a)

rate = 20 students per hour

time between 11:30 am and 11:45 am = 15 min

X follow poisson distribution ,

lambda = 20/60 * 15 =5 , support = {0,1,2,...}

P(X >= 1) = 1 - P(X = 0) =   1 - e^(-5) = 0.993262053

b)

P(X < 4 |X>= 1)

= P(1<= X < 4) /P(X>=)

= P(1 <= X<=3) /P(X >=1)

= 0.258288/0.993262

= 0.26004

c)

11:45 to 12:15 is 30 min

hence lambda = 10

required probability =

P(X = 3 with lambda = 5) * P(Y = 8 with lambda = 10)

= 0.140374 * 0.112599

=

0.015806

d)

E(X) = 3.5 * 20 /hour

= 70