1. **State the problem:** We have a probability distribution with values of $x$ and their probabilities $P(X)$. We need to complete the table by calculating $x \cdot P(X)$, $x - \mu$, $(x - \mu)^2$, and $(x - \mu)^2 \cdot P(X)$ for each $x$. Then find the mean ($\mu$), variance ($\sigma^2$), and standard deviation ($\sigma$), rounding mean and standard deviation to the nearest hundredths. 2. **Formulae and rules:** - Mean (expected value): $$\mu = \sum x P(X)$$ - Variance: $$\sigma^2 = \sum (x - \mu)^2 P(X)$$ - Standard deviation: $$\sigma = \sqrt{\sigma^2}$$ 3. **Calculate $x \cdot P(X)$:** - For $x=1$: $1 \times 0.2 = 0.2$ - For $x=4$: $4 \times 0.4 = 1.6$ - For $x=5$: $5 \times 0.2 = 1.0$ - For $x=7$: $7 \times 0.2 = 1.4$ Sum of $x \cdot P(X)$: $$0.2 + 1.6 + 1.0 + 1.4 = 4.2$$ So, mean $$\mu = 4.2$$ 4. **Calculate $x - \mu$ for each $x$:** - $1 - 4.2 = -3.2$ - $4 - 4.2 = -0.2$ - $5 - 4.2 = 0.8$ - $7 - 4.2 = 2.8$ 5. **Calculate $(x - \mu)^2$ for each $x$:** - $(-3.2)^2 = 10.24$ - $(-0.2)^2 = 0.04$ - $(0.8)^2 = 0.64$ - $(2.8)^2 = 7.84$ 6. **Calculate $(x - \mu)^2 \cdot P(X)$ for each $x$:** - $10.24 \times 0.2 = 2.048$ - $0.04 \times 0.4 = 0.016$ - $0.64 \times 0.2 = 0.128$ - $7.84 \times 0.2 = 1.568$ Sum of $(x - \mu)^2 \cdot P(X)$: $$2.048 + 0.016 + 0.128 + 1.568 = 3.76$$ So, variance $$\sigma^2 = 3.76$$ 7. **Calculate standard deviation:** $$\sigma = \sqrt{3.76} \approx 1.94$$ 8. **Interpretation:** The mean $\mu = 4.2$ represents the average value of the random variable $X$. The standard deviation $\sigma = 1.94$ indicates the average amount by which values of $X$ deviate from the mean. A standard deviation of about 1.94 shows moderate spread around the mean in this distribution. **Completed table:** | x | P(X) | x \cdot P(X) | $x - \mu$ | $(x - \mu)^2$ | $(x - \mu)^2 \cdot P(X)$ | |---|-------|--------------|-----------|---------------|---------------------------| | 1 | 0.2 | 0.2 | -3.2 | 10.24 | 2.048 | | 4 | 0.4 | 1.6 | -0.2 | 0.04 | 0.016 | | 5 | 0.2 | 1.0 | 0.8 | 0.64 | 0.128 | | 7 | 0.2 | 1.4 | 2.8 | 7.84 | 1.568 | **Final answers:** - Mean $\mu = 4.2$ - Variance $\sigma^2 = 3.76$ - Standard deviation $\sigma = 1.94$