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

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SUMMARY

The discussion focuses on calculating the Probability Mass Function (PMF) for the number of modems in use by an internet service provider serving 1000 customers with 50 modems. Each customer has a 0.01 probability of needing a connection independently. The PMF is derived in two parts: first using a binomial distribution, and then approximating it with a Poisson distribution for easier calculations. The probability of having more customers needing a connection than available modems is also analyzed, providing both exact and approximate formulas based on the Poisson approximation.

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wooya
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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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