Transition from Discrete to Continuous

Poisson random variable is an important random variable as it enables the transition from discrete to continuous random variables for some distribution. In the question below, you will need to know what distribution you need to use. You also need to be extra careful for the dimension used for the intervals.

In a base station next to Medipol, on average $250$ telephone calls per hour arrive. For the time being we assume unlimited capacity for the base station. If the incoming calls follow Poisson distribution, $c4=12$

$($a$)(5$ Pts$.)$ For the duration of $c_4$ minutes, what is the probability that we have $100$ users connected? Comment on the result.

$($b$)(5$ Pts$.)$ What is the probability that we get $c_{4}1000$ calls in one day? Comment on the result.

$($c$)(5$ Pts$.)$ If $x$ represents the number of calls per hour, determine

$E[e^x+x^2]$

$($d$)(5$ Pts$.)$ What is the probability that we get the first user connected to the system after $2c_4$ seconds? Comment on the result.