Customers arrive at coffee shop according to a Poisson process at a mean rate of 5 customers every 10 minutes. (a) Let X

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Customers arrive at coffee shop according to a Poisson process
at a mean rate of 5 customers every 10 minutes.
(a) Let X denote the waiting time (in minutes, counted from 7
am) until the first customer arrives. What is the pdf of X? Name
the distribution if it has a name.
(b) Find the probability that the owner of the coffee shop will
have to wait longer than 5 minutes before the arrival of the first
customer.
(c) Find the conditional probability that the owner of the
coffee shop will have to wait longer than 5 minutes before the
arrival of the first customer, given that no customer came in the
first three minutes.
(d) What is the probability that the owner of the coffee shop
will have to wait longer than 28.41 minutes before the arrival of
the 10th customer?
Use two different approaches to solve this question by using the
relationship be- tween the gamma, Poisson, and chi-square
distributions. You are allowed to use R to obtain the numerical
answers.