1. **Stating the problem:** We are given a table with values of a random variable $x$ and their probabilities $p(x)$, and we want to calculate the variance and standard deviation. 2. **Formula for variance:** The variance $\sigma^2$ of a discrete random variable is given by $$\sigma^2 = E[(X - \mu)^2] = \sum p(x)(x - \mu)^2$$ where $\mu = E[X] = \sum p(x)x$ is the mean. 3. **Example table:** Suppose we have | $x$ | 1 | 2 | 3 | 4 | |-----|---|---|---|---| | $p(x)$ | 0.1 | 0.3 | 0.4 | 0.2 | 4. **Calculate the mean $\mu$:** $$\mu = 0.1 \times 1 + 0.3 \times 2 + 0.4 \times 3 + 0.2 \times 4 = 0.1 + 0.6 + 1.2 + 0.8 = 2.7$$ 5. **Calculate variance $\sigma^2$:** $$\sigma^2 = 0.1(1 - 2.7)^2 + 0.3(2 - 2.7)^2 + 0.4(3 - 2.7)^2 + 0.2(4 - 2.7)^2$$ Calculate each term: $$0.1 \times ( -1.7)^2 = 0.1 \times 2.89 = 0.289$$ $$0.3 \times ( -0.7)^2 = 0.3 \times 0.49 = 0.147$$ $$0.4 \times 0.3^2 = 0.4 \times 0.09 = 0.036$$ $$0.2 \times 1.3^2 = 0.2 \times 1.69 = 0.338$$ Sum these: $$\sigma^2 = 0.289 + 0.147 + 0.036 + 0.338 = 0.81$$ 6. **Calculate standard deviation $\sigma$:** $$\sigma = \sqrt{0.81} = 0.9$$ **Final answer:** Variance is $0.81$ and standard deviation is $0.9$.