Questions: Dirty air: The federal government has enacted maximum allowable standards for air pollutants such as ozone. Let X be the number of days per year that the level of air pollution exceeds the standard in a certain city. The probability distribution of X is given by x 0 1 2 3 4 P(x) 0.33 0.39 0.19 0.04 0.05

Dirty air: The federal government has enacted maximum allowable standards for air pollutants such as ozone. Let X be the number of days per year that the level of air pollution exceeds the standard in a certain city. The probability distribution of X is given by

x 0 1 2 3 4
P(x) 0.33 0.39 0.19 0.04 0.05

Transcript text: Dirty air: The federal government has enacted maximum allowable standards for air pollutants such as ozone. Let $X$ be the number of days per year that the level of air pollution exceeds the standard in a certain city. The probability distribution of $X$ is given by \begin{tabular}{c|ccccc} $x$ & 0 & 1 & 2 & 3 & 4 \\ \hline$P(x)$ & 0.33 & 0.39 & 0.19 & 0.04 & 0.05 \end{tabular}

Solution

Solution Steps

Step 1: Computing the Mean ($\mu_X$)

The mean ($\mu_X$) is computed using the formula: \[ \mu_X = \sum_{i} x_i P(x_i) \] Substituting the given values, we get $\mu_X = 1.09$.

Step 2: Computing the Standard Deviation ($\sigma_X$)

The standard deviation ($\sigma_X$) is computed using the formula: \[ \sigma_X = \sqrt{\sum_{i} (x_i - \mu_X)^2 P(x_i)} \] Substituting the given values and the computed mean, we get $\sigma_X = 1.06$.

Step 3: Computing Probabilities of Specific Outcomes

The probability of specific outcomes is computed by summing the probabilities of the desired outcomes. For the given specific outcomes, the total probability is 0.28.