1. **Problem Statement:** Find the mean, variance, and standard deviation of the probability distribution for the number of heads $X$ when two coins are tossed. 2. **Given Data:** Number of heads $X$: 0, 1, 2 Probabilities $P(X)$: $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ 3. **Mean (Expected Value) Formula:** $$\mu = E(X) = \sum x_i P(x_i)$$ 4. **Calculate Mean:** $$E(X) = 0 \times \frac{1}{4} + 1 \times \frac{1}{2} + 2 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{2}{4} = \frac{1}{2} + \frac{1}{2} = 1$$ 5. **Variance Formula:** $$\sigma^2 = Var(X) = E(X^2) - [E(X)]^2$$ 6. **Calculate $E(X^2)$:** $$E(X^2) = 0^2 \times \frac{1}{4} + 1^2 \times \frac{1}{2} + 2^2 \times \frac{1}{4} = 0 + \frac{1}{2} + \frac{4}{4} = \frac{1}{2} + 1 = \frac{3}{2}$$ 7. **Calculate Variance:** $$Var(X) = \frac{3}{2} - (1)^2 = \frac{3}{2} - 1 = \frac{1}{2}$$ 8. **Standard Deviation Formula:** $$\sigma = \sqrt{Var(X)}$$ 9. **Calculate Standard Deviation:** $$\sigma = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}$$ **Final answers:** - Mean $\mu = 1$ - Variance $\sigma^2 = \frac{1}{2}$ - Standard deviation $\sigma = \frac{\sqrt{2}}{2}$