1. **Problem statement:** We have a discrete random variable $X$ with values $k=0,1,2,3$ and probabilities $P(X=k) = p_0, p_1, 0.2, 0.1$ respectively. Given that the expectation $E(X) = 1$, we need to find $p_0$ and $p_1$, and then calculate the variance $\sigma^2$ of $X$. 2. **Formulas and rules:** - The sum of all probabilities must be 1: $$p_0 + p_1 + 0.2 + 0.1 = 1$$ - The expectation is: $$E(X) = \sum k P(X=k) = 0 \cdot p_0 + 1 \cdot p_1 + 2 \cdot 0.2 + 3 \cdot 0.1 = 1$$ - Variance formula: $$\sigma^2 = E(X^2) - (E(X))^2$$ where $$E(X^2) = \sum k^2 P(X=k)$$ 3. **Find $p_0$ and $p_1$:** - From the sum of probabilities: $$p_0 + p_1 + 0.2 + 0.1 = 1 \Rightarrow p_0 + p_1 = 0.7$$ - From the expectation: $$0 \cdot p_0 + 1 \cdot p_1 + 2 \cdot 0.2 + 3 \cdot 0.1 = 1$$ $$p_1 + 0.4 + 0.3 = 1$$ $$p_1 + 0.7 = 1 \Rightarrow p_1 = 0.3$$ - Substitute $p_1$ back: $$p_0 + 0.3 = 0.7 \Rightarrow p_0 = 0.4$$ 4. **Calculate variance:** - Compute $E(X^2)$: $$E(X^2) = 0^2 \cdot 0.4 + 1^2 \cdot 0.3 + 2^2 \cdot 0.2 + 3^2 \cdot 0.1 = 0 + 0.3 + 4 \cdot 0.2 + 9 \cdot 0.1$$ $$= 0.3 + 0.8 + 0.9 = 2.0$$ - Variance: $$\sigma^2 = E(X^2) - (E(X))^2 = 2.0 - 1^2 = 2.0 - 1 = 1.0$$ --- **Final answers:** - $p_0 = 0.4$ - $p_1 = 0.3$ - Variance $\sigma^2 = 1.0$