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A factory produces light bulbs. Historical data shows that \(2\pourcent\) of the bulbs are defective. A quality control manager picks a sample of 50 bulbs to inspect.
Let \(X_k\) be the random variable representing the state of the \(k\)-th bulb selected.
  1. Define the variable \(X_k\) to model the proportion of defective bulbs.
  2. Give the mean and standard deviation of a single variable \(X_k\).
  3. Express the sample proportion \(\hat{P}\) in terms of \(X_k\).

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