1. **Stating the problem:** We have a discrete probability distribution with values $x$, probabilities $P(x)$, and related calculations including mean $\mu$, sample mean $\bar{X}$, deviations, and weighted squared deviations. 2. **Given data:** - $x = \{9, 11, 17, 8, 11\}$ - $P(x)$ values sum to 1. - Mean $\mu = 12.071428$ - Sample mean $\bar{X} = 11.2$ 3. **Formulas used:** - Mean (expected value): $$\mu = \sum x P(x)$$ - Sample mean: $$\bar{X} = \frac{\sum x}{n}$$ - Deviation from sample mean: $$x - \bar{X}$$ - Squared deviation: $$(x - \bar{X})^2$$ - Weighted squared deviation from mean: $$(x - \mu)^2 P(x)$$ 4. **Explanation:** - The mean $\mu$ is the expected value of the distribution, calculated by summing each $x$ multiplied by its probability. - The sample mean $\bar{X}$ is the average of the $x$ values. - Deviations measure how far each $x$ is from the mean or sample mean. - Squared deviations are used to calculate variance. - Weighted squared deviations $(x - \mu)^2 P(x)$ contribute to the variance of the distribution. 5. **Intermediate calculations:** - Sum of $P(x)$ is 1, confirming a valid probability distribution. - $\mu = 12.071428$ is given. - $\bar{X} = 11.2$ is given. - Deviations and squared deviations are calculated as shown in the table. 6. **Final answer:** - The mean of the distribution is $\mu = 12.071428$. - The sample mean is $\bar{X} = 11.2$. - The table correctly shows deviations and weighted squared deviations used for variance calculations. This completes the analysis of the given probability distribution data.