Comme un jeu

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

You have invited \(n=64\) guests to a party. You need to make sandwiches for the guests.You believe that the number of sandwiches a single guest needs, \(X_i\), takes values \(0, 1,\) or \(2\) with probabilities \(\frac{1}{4}, \frac{1}{2}\), and \(\frac{1}{4}\) respectively.
Assume that the number of sandwiches each guest needs is independent of the others.Let \(S_{64}\) be the total number of sandwiches needed.

Find the mean \(\mu\) and the variance \(\sigma^2\) of the number of sandwiches for a single guest.
Find the expected total number of sandwiches \(E(S_{64})\) and the standard deviation of the total \(\sigma(S_{64})\).
Can we assume \(S_{64}\) is normally distributed? Explain.
How many sandwiches should you make so that you are \(95 \pourcent\) sure that there is no shortage (i.e., find \(k\) such that \(P(S_{64} \le k) \ge 0.95\))?

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