02 May 1. You must work out your answers for this question by hand (i.e. do
1. You must work out your answers for this question by hand (i.e. do not use R or another computer program but
you can use a scientific calculator to aid you with basic calculations when necessary).
Suppose that in a mass production line, 0.2% of the integrated circuit chips are defective. To maintain and
monitor the quality of the chips, a sample of 800 chips is selected at regular intervals for inspection. If X denote
the number of defective chips found in the sample, we can write: X ~ BIN(800, 0.002).
(a) What is E(X) and Var(X)?
(b) Using the probability mass function for a binomial random variable, what is the probability that there are
no ’defective’ chips? That is, what is P(X = 0)? Hint: Do not, for example, try and compute the factorial
of 800 when working out
800
0
. Look for simplifications first.
(c) What is the probability that there is exactly one ’defective’ chip? Hint: similar to the hint for (b).
(d) Suppose to introduce a new production line, the manager requires at least two ’defective’ chips, out of the
800 chips. What is the probability of this occurring?
(e) The Poisson distribution provides an excellent approximation to the Binomial distribution when p is small
and n is large.1 Using the fact that the POIS(np) distribution can be used to approximate the BIN(n, p)
distribution for large n and small p, use the Poisson distribution to approximate the probabilities you
calculated in (b), (c) and (d). Were the approximations close?
2. Suppose that cars arrive at a fast-food drive through window according to a Poisson process at a rate of 6 cars
per hour. There is a single point of service who can process 10 orders per hour. If the operator at the window
is busy processing an existing order, any arriving customers will wait in a queue and join the end of the queue
behind any other cars already waiting. You may assume that the system can be modeled as an M|M|1 queue.
Answer the following questions.
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