Telephone calls arrive at a doctor’s office according to a poisson process on the average of two every 3 minutes. let x denote the waiting time until the first call that arrives after 10 a. m.
(a) what is the pdf of x?
(b) find p(x > 2).

a) when

otherwise

b)

Step-by-step explanation:

First of all we have a Poisson process with a mean equal to :

μ = λ = (Two phone calls every 3 minutes)

Let's define the random variable X.

X : ''The waiting time until the first call that arrives after 10 a.m.''

a) The waiting time between successes of a Poisson process is modeled with a exponential distribution :

X ~ ε (λ)    Where λ is the mean of the Poisson process

The exponential distribution follows the next probability density function :

I replace λ = a for the equation.

With

and

Otherwise

In this exercise λ= a = ⇒

Otherwise

That's incise a)

For b) We must integrate between 2 and ∞ to obtain the probability or either use the cumulative probability function of the exponential

when

and

when

For this exercise

Therefore

it is a. 4.2 because 7.6-3.4 gets the remaining length of 4.2. i might be wrong but i am pretty sure this is the answer.