Expected Value Stddev 847637

1. **State the problem:** We are given a discrete probability distribution with values of $x$ and their corresponding probabilities $P(x)$. We need to find:\n\na. The expected value $E(X)$ of the distribution.\nb. The standard deviation $\sigma$ of the distribution.\n\n2. **Recall formulas:**\n- Expected value: $$E(X) = \sum x_i P(x_i)$$\n- Variance: $$\sigma^2 = \sum (x_i - E(X))^2 P(x_i)$$\n- Standard deviation: $$\sigma = \sqrt{\sigma^2}$$\n\n3. **Calculate the expected value:**\n$$E(X) = 0 \times 0.15 + 1 \times 0.1 + 2 \times 0.1 + 3 \times 0.65$$\n$$= 0 + 0.1 + 0.2 + 1.95 = 2.25$$\n\n4. **Calculate the variance:**\nCalculate each squared difference weighted by probability:\n$$ (0 - 2.25)^2 \times 0.15 = ( -2.25)^2 \times 0.15 = 5.0625 \times 0.15 = 0.759375 $$\n$$ (1 - 2.25)^2 \times 0.1 = ( -1.25)^2 \times 0.1 = 1.5625 \times 0.1 = 0.15625 $$\n$$ (2 - 2.25)^2 \times 0.1 = ( -0.25)^2 \times 0.1 = 0.0625 \times 0.1 = 0.00625 $$\n$$ (3 - 2.25)^2 \times 0.65 = (0.75)^2 \times 0.65 = 0.5625 \times 0.65 = 0.365625 $$\nSum these to get variance:\n$$\sigma^2 = 0.759375 + 0.15625 + 0.00625 + 0.365625 = 1.2875$$\n\n5. **Calculate the standard deviation:**\n$$\sigma = \sqrt{1.2875} \approx 1.135\ \Rightarrow\ 1.14 \text{ (rounded to two decimals)}$$\n\n**Final answers:**\n- Expected value $E(X) = 2.25$\n- Standard deviation $\sigma = 1.14$