Answer:
The probability that 5 messages are received in 1 hour is

The probability that 10 messages are received in 1.5 hours is

The probability that less than 2 messages are received in 1/2 hour is

Explanation:
The probability distribution of a Poisson random variable representing the number of successes occurring in a given time interval or a specified region of space is given by the formula:
where
mean number of successes in the given time interval or region of space.

a) What is the probability that 5 messages are received in 1 hour?
From the information given we know that

Applying the Poisson random variable formula we get:

b) What is the probability that 10 messages are received in 1.5 hours?
We know from the information given that in 1 hour, 5 messages are received. In 0.5 hours we received 2.5 messages therefore in 1.5 hours we received 7.5 messages

Applying the Poisson random variable formula we get:

c) What is the probability that less than 2 messages are received in 1/2 hour?
We know from the information given that in 1 hour, 5 messages are received therefore in 0.5 hours we received 2.5 messages

To find the probability that less than 2 messages are received in 1/2 hour we find the values when
and
because

Applying the Poisson random variable formula we get:


