Question

Let the mean success rate of a Poisson process be 12 successes per hour.

a. Find the expected number of successes in a 19 minutes period. (Round your answer to 4 decimal places.)

b. Find the probability of at least 2 successes in a given 19 minutes period. (Round your answer to 4 decimal places.)

c. Find the expected number of successes in a two hours 30 minutes period. (Round your answer to 2 decimal places.)

d. Find the probability of 23 successes in a given two hours 30 minutes period. (Do not round intermediate calculations. Round your final answer to 4 decimal places.)

Answer 1

X ~ Poisson ()

Where = 12 successes per hour.

Poisson probability distribution is

P(X) = e^-X / X!

a)

E(X) = 12* 19 / 60 = 3.8

b)

P(X >= 2) = 1 - P(X <= 1)

= 1 - [ P(X = 0) + P(X = 1) ]

= 1 - [e^-3.8 + e^-3.8 * 3.8 ]

= 0.8926

c)

In 2 hour 30 minutes, there are 2 * 60 + 30 = 150 minutes.

E(X) = 12 * 150 / 60 = 30

d)

P(X = 23) = e^-30 * 30²³ / 23!

= 0.0341