Comme un jeu

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

You are managing a moving company. You have \(n=100\) students moving out of a dormitory today. You need to provide boxes.
Based on past data, the number of boxes a single student needs, \(X_i\), takes values \(2, 3,\) or \(4\) with probabilities \(0.2, 0.5\), and \(0.3\) respectively.
Assume that the number of boxes each student needs is independent of the others.
Let \(S_{100}\) be the total number of boxes needed.

Find the mean \(\mu\) and the variance \(\sigma^2\) of the number of boxes for a single student.
Find the expected total number of boxes \(E(S_{100})\) and the standard deviation of the total \(\sigma(S_{100})\).
Can we assume \(S_{100}\) is normally distributed? Explain.
How many boxes should you bring so that you are \(99 \pourcent\) sure that there is no shortage (i.e., find \(k\) such that \(P(S_{100} \le k) \ge 0.99\))?

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