1. **Problem statement:** Given a random variable $X$ with values and probabilities: | $x$ | 0 | 10 | 20 | 30 | |---|---|----|----|----| | $P(X=x)$ | 0.10 | 0.05 | 0.80 | 0.05 | Find (a) the expectation $E(X)$ and (b) the variance $\mathrm{Var}(X)$. 2. **Formulas:** - Expectation: $$E(X) = \sum x_i P(X=x_i)$$ - Variance: $$\mathrm{Var}(X) = E(X^2) - [E(X)]^2$$ where $$E(X^2) = \sum x_i^2 P(X=x_i)$$ 3. **Calculate expectation $E(X)$:** $$E(X) = 0 \times 0.10 + 10 \times 0.05 + 20 \times 0.80 + 30 \times 0.05$$ $$= 0 + 0.5 + 16 + 1.5 = 18$$ 4. **Calculate $E(X^2)$:** $$E(X^2) = 0^2 \times 0.10 + 10^2 \times 0.05 + 20^2 \times 0.80 + 30^2 \times 0.05$$ $$= 0 + 100 \times 0.05 + 400 \times 0.80 + 900 \times 0.05$$ $$= 0 + 5 + 320 + 45 = 370$$ 5. **Calculate variance $\mathrm{Var}(X)$:** $$\mathrm{Var}(X) = E(X^2) - [E(X)]^2 = 370 - 18^2 = 370 - 324 = 46$$ **Final answers:** - (a) $E(X) = 18$ - (b) $\mathrm{Var}(X) = 46$