1. **State the problem:** Find the variance and standard deviation of a given probability distribution with values $x$ and probabilities $P(X)$. 2. **Recall formulas:** - Mean (expected value): $$\mu = \sum x_i P(x_i)$$ - Variance: $$\sigma^2 = \sum (x_i - \mu)^2 P(x_i) = \sum x_i^2 P(x_i) - \mu^2$$ - Standard deviation: $$\sigma = \sqrt{\sigma^2}$$ 3. **Given:** The problem mentions $x$, $P(X)$, and $x^2 P(X)$ but does not provide explicit values. To find variance, we need: - $\mu = \sum x_i P(x_i)$ - $\sum x_i^2 P(x_i)$ 4. **Calculate mean:** $$\mu = \sum x_i P(x_i)$$ 5. **Calculate variance:** $$\sigma^2 = \sum x_i^2 P(x_i) - \mu^2$$ 6. **Calculate standard deviation:** $$\sigma = \sqrt{\sigma^2}$$ **Note:** Without explicit values for $x_i$ and $P(x_i)$, we cannot compute numeric answers. Please provide the values of $x$ and $P(X)$ to proceed.