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Probability (0 to 1): Mean (μ): Standard Deviation (σ):

An Internet service provider sets up a local access point which serves \(n=5000\) subscribers.
At any given time, each subscriber has a probability of \(p=0.2\) of being connected. We assume that the behavior of each subscriber is independent of the others.
Let \(S_{5000}\) be the total number of active connections at a given time.

Find the expected number of active connections, \(E(S_{5000})\).
Find the standard deviation of the number of active connections, \(\sigma(S_{5000})\).
Using the Central Limit Theorem, approximate the distribution of \(S_{5000}\).
Find the minimum number of simultaneous connections, \(k\), that the access point must be able to handle so that the probability of the system **not** being saturated is greater than \(99.99\pourcent\) (i.e., \(P(S_{5000} \le k) > 0.9999\)).

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