1. **State the problem:** We are given a discrete random variable with scores and their corresponding probabilities. We need to find the expected value (mean) and the standard deviation of this random variable. 2. **Formulas:** - Expected value (mean) $E(X)$ is given by: $$E(X) = \sum p_i x_i$$ where $p_i$ are probabilities and $x_i$ are scores. - Variance $\sigma^2$ is given by: $$\sigma^2 = E(X^2) - [E(X)]^2$$ where $$E(X^2) = \sum p_i x_i^2$$ - Standard deviation $\sigma$ is the square root of variance: $$\sigma = \sqrt{\sigma^2}$$ 3. **Calculate expected value:** $$E(X) = (0.16)(1) + (0.09)(6) + (0.13)(11) + (0.22)(12) + (0.15)(13) + (0.25)(14)$$ $$= 0.16 + 0.54 + 1.43 + 2.64 + 1.95 + 3.5 = 10.22$$ 4. **Calculate $E(X^2)$:** $$E(X^2) = (0.16)(1^2) + (0.09)(6^2) + (0.13)(11^2) + (0.22)(12^2) + (0.15)(13^2) + (0.25)(14^2)$$ $$= (0.16)(1) + (0.09)(36) + (0.13)(121) + (0.22)(144) + (0.15)(169) + (0.25)(196)$$ $$= 0.16 + 3.24 + 15.73 + 31.68 + 25.35 + 49 = 125.16$$ 5. **Calculate variance:** $$\sigma^2 = E(X^2) - [E(X)]^2 = 125.16 - (10.22)^2 = 125.16 - 104.45 = 20.71$$ 6. **Calculate standard deviation:** $$\sigma = \sqrt{20.71} \approx 4.55$$ **Final answers:** - Expected value $E(X) = 10.22$ - Standard deviation $\sigma \approx 4.55$