To answer your questions regarding the random variable X, which represents the number of days per year that the level of air pollution exceeds the federal standards, we need the specific probability distribution for X (such as a table or a formula).
Since you have not provided the probability distribution, I will provide a general approach you can take depending on what type of distribution it is (e.g., Binomial, Poisson):
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Find P(1) - This requires finding the probability that X = 1. Using the probability mass function (PMF) from your distribution, look for the value corresponding to X = 1.
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Find P(3 or fewer) - This requires calculating the cumulative probability for X values 0, 1, 2, and 3: \[ P(X \leq 3) = P(0) + P(1) + P(2) + P(3) \]
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Find the probability that the standard is exceeded on at least three days: \[ P(X \geq 3) = 1 - P(X \leq 2) \] or directly calculate \( P(3) + P(4) + \ldots \).
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Find the probability that the standard is exceeded on more than one day: \[ P(X > 1) = 1 - P(X \leq 1) = 1 - [P(0) + P(1)] \]
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Compute the mean \( \mu_X \) - The mean or expected value of a discrete random variable can be computed using: \[ \mu_X = \sum_{x=0}^{n} x \cdot P(x) \] Here, "n" is the maximum value of X and the sums are computed for each value of X with their respective probabilities.
To get specific values for each of the above steps, you will need to provide the probability distribution of X. Please share that information to enable detailed computations!