1. **State the problem:** We are given a probability distribution of the number of quarantine protocol violators and need to find the mean, variance, and standard deviation. 2. **Recall formulas:** - Mean (Expected value): $$\mu = E(X) = \sum x_i P(x_i)$$ - Variance: $$\sigma^2 = Var(X) = E(X^2) - [E(X)]^2$$ - Standard deviation: $$\sigma = \sqrt{Var(X)}$$ 3. **Calculate the mean:** $$\mu = 25 \times \frac{3}{20} + 30 \times \frac{9}{20} + 35 \times \frac{7}{20} + 40 \times \frac{1}{20}$$ Calculate each term: $$25 \times \frac{3}{20} = \frac{75}{20} = 3.75$$ $$30 \times \frac{9}{20} = \frac{270}{20} = 13.5$$ $$35 \times \frac{7}{20} = \frac{245}{20} = 12.25$$ $$40 \times \frac{1}{20} = \frac{40}{20} = 2$$ Sum these: $$\mu = 3.75 + 13.5 + 12.25 + 2 = 31.5$$ 4. **Calculate $E(X^2)$:** $$E(X^2) = 25^2 \times \frac{3}{20} + 30^2 \times \frac{9}{20} + 35^2 \times \frac{7}{20} + 40^2 \times \frac{1}{20}$$ Calculate each term: $$25^2 = 625, \quad 625 \times \frac{3}{20} = \frac{1875}{20} = 93.75$$ $$30^2 = 900, \quad 900 \times \frac{9}{20} = \frac{8100}{20} = 405$$ $$35^2 = 1225, \quad 1225 \times \frac{7}{20} = \frac{8575}{20} = 428.75$$ $$40^2 = 1600, \quad 1600 \times \frac{1}{20} = \frac{1600}{20} = 80$$ Sum these: $$E(X^2) = 93.75 + 405 + 428.75 + 80 = 1007.5$$ 5. **Calculate variance:** $$Var(X) = E(X^2) - (E(X))^2 = 1007.5 - (31.5)^2$$ Calculate square: $$31.5^2 = 992.25$$ Subtract: $$Var(X) = 1007.5 - 992.25 = 15.25$$ 6. **Calculate standard deviation:** $$\sigma = \sqrt{15.25} \approx 3.905$$ **Final answers:** - Mean: $31.5$ - Variance: $15.25$ - Standard deviation: $3.905$