What is the PMF of the number of modems in use at the given time?

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The discussion focuses on determining the probability mass function (PMF) for the number of modems in use by an internet service provider serving 1000 customers with a 0.01 connection probability. The PMF can be calculated using the binomial distribution for part (a). In part (b), the PMF is approximated using the Poisson distribution, which simplifies calculations by relying on the expected number of connections. The probability of more customers needing a connection than available modems is also analyzed, providing both exact and approximate formulas. The Poisson distribution is highlighted for its ease of use in these calculations.
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An internet service provider uses 50 modems to serve the needs of 1000 customers. It is estimated that at a given time, each customer will need a connection with probability 0.01 independently of the other customers.
(a) What is the PMF of the number of modems in use at the given time?
(b) Repeat part (a) by approximating the PMF of the number of customers that need a connection with a Poisson PMF.
(c) What is the probability that there are more customers needing a connection than there
are modems? Provide an exact, as well as an approximate formula based on the Poisson
approximation of part (b).
 
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What's the problem? The Poisson distribution is really easy to use: just find the expected number of people who will connect at a given point, then use the probability mass formula.
 

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