1. **State the problem:** We have a random variable $X$ with values $0, 1, 2, 3$ and corresponding probabilities $0.1, 0.4, 0.3, 0.2$. We need to find the mean (expected value) $\mu$ of $X$. 2. **Formula:** The mean of a discrete random variable is given by $$\mu = E(X) = \sum x_i p_i$$ where $x_i$ are the values and $p_i$ are their probabilities. 3. **Calculate:** $$\mu = 0 \times 0.1 + 1 \times 0.4 + 2 \times 0.3 + 3 \times 0.2$$ $$= 0 + 0.4 + 0.6 + 0.6$$ $$= 1.6$$ 4. **Interpretation:** The mean $\mu = 1.6$ represents the average value of the random variable $X$ weighted by their probabilities. **Final answer:** $$\boxed{1.6}$$