Question

The number of buses arriving at the bus stop for T minutes is defined as a random variable B. The average (expected value) of random variable B is T / 5.

(1)A value indicating the average number of occurrences per unit time in the Poisson distribution. What is the average rate of arrival per second?

(2)find PMF of B

(3)Find the probability of 3 buses arriving in 2 minutes

(4)Find the probability that the bus will not arrive in 10 minutes

(5)Find the probability of at least one bus arriving in T minutes

Answer 1

1. average number of occurrences per unit time = B/T

= (T/5) / T

= 1/5 = 0.2

2.

P(B buses in T time) = e^(-0.2*T) * (0.2*T)^B / (B!)

3.

P(B buses in T time) = e^(-0.2*T) * (0.2*T)^B / (B!)

P(3 buses in 2 min) = e^(-0.2*2) * (0.2*2)^3 / (3!)

= 0.0072

4.

P(B buses in T time) = e^(-0.2*T) * (0.2*T)^B / (B!)

P(0 buses in 10 min) = e^(-0.2*10) * (0.2*10)^0 / (0!)

= 0.1353

5.

probability of at least one bus arriving in T minutes

P(B >=1 buses in T time) = 1 - P(0 buses in T min)

= 1 - e^(-0.2*T) * (0.2*T)^0 / (0!)

probability of at least one bus arriving in T minutes

= 1 - e^(-0.2*T) * (0.2*T)^0 / (0!)

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