1. **State the problem:** We are given the probability distribution of expected returns for Stock X and need to calculate the standard deviation and coefficient of variation of these returns. 2. **Given data:** - Probabilities: $p = [0.1, 0.2, 0.4, 0.2, 0.1]$ - Returns for Stock X: $r = [-0.10, 0.02, 0.12, 0.20, 0.38]$ - Expected return $E(R) = 0.12$ 3. **Formulas:** - Variance: $$\sigma^2 = \sum p_i (r_i - E(R))^2$$ - Standard deviation: $$\sigma = \sqrt{\sigma^2}$$ - Coefficient of variation: $$CV = \frac{\sigma}{E(R)}$$ 4. **Calculate variance:** Calculate each squared deviation weighted by probability: $$0.1 \times (-0.10 - 0.12)^2 = 0.1 \times (-0.22)^2 = 0.1 \times 0.0484 = 0.00484$$ $$0.2 \times (0.02 - 0.12)^2 = 0.2 \times (-0.10)^2 = 0.2 \times 0.01 = 0.002$$ $$0.4 \times (0.12 - 0.12)^2 = 0.4 \times 0^2 = 0$$ $$0.2 \times (0.20 - 0.12)^2 = 0.2 \times 0.08^2 = 0.2 \times 0.0064 = 0.00128$$ $$0.1 \times (0.38 - 0.12)^2 = 0.1 \times 0.26^2 = 0.1 \times 0.0676 = 0.00676$$ Sum these to get variance: $$\sigma^2 = 0.00484 + 0.002 + 0 + 0.00128 + 0.00676 = 0.01488$$ 5. **Calculate standard deviation:** $$\sigma = \sqrt{0.01488} \approx 0.122$$ 6. **Calculate coefficient of variation:** $$CV = \frac{0.122}{0.12} \approx 1.02$$ **Final answers:** - Standard deviation of expected returns for Stock X is approximately $0.122$ (or 12.2%). - Coefficient of variation for Stock X is approximately $1.02$.