1. **State the problem:** We have a probability distribution with values of $k$ and their probabilities $P(k)$: | $k$ | $P(k)$ | |-----|--------| | 1 | 0.2 | | 4 | 0.4 | | 5 | 0.2 | | 7 | 0.2 | We need to complete the table, find the mean ($p$), variance, and standard deviation, then interpret the results. 2. **Formula for mean (expected value):** $$p = \sum k \cdot P(k)$$ 3. **Calculate $k \cdot P(k)$ for each $k$:** - For $k=1$: $1 \times 0.2 = 0.2$ - For $k=4$: $4 \times 0.4 = 1.6$ - For $k=5$: $5 \times 0.2 = 1.0$ - For $k=7$: $7 \times 0.2 = 1.4$ Sum these to get the mean: $$p = 0.2 + 1.6 + 1.0 + 1.4 = 4.2$$ 4. **Calculate $x - p$ for each $k$:** - $1 - 4.2 = -3.2$ - $4 - 4.2 = -0.2$ - $5 - 4.2 = 0.8$ - $7 - 4.2 = 2.8$ 5. **Calculate $(x - p)^2$ for each $k$:** - $(-3.2)^2 = 10.24$ - $(-0.2)^2 = 0.04$ - $(0.8)^2 = 0.64$ - $(2.8)^2 = 7.84$ 6. **Calculate $(x - p)^2 \times P(k)$ for each $k$:** - $10.24 \times 0.2 = 2.048$ - $0.04 \times 0.4 = 0.016$ - $0.64 \times 0.2 = 0.128$ - $7.84 \times 0.2 = 1.568$ Sum these to get the variance: $$\text{Variance} = 2.048 + 0.016 + 0.128 + 1.568 = 3.76$$ 7. **Calculate the standard deviation:** $$\text{Standard deviation} = \sqrt{3.76} \approx 1.94$$ 8. **Complete the table:** | $k$ | $P(k)$ | $k \cdot P(k)$ | $x - p$ | $(x - p)^2$ | $(x - p)^2 \times P(k)$ | |-----|--------|-----------------|---------|-------------|--------------------------| | 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 | 9. **Interpretation:** The mean value of 4.2 represents the expected value of $k$ in this distribution. The variance of 3.76 and standard deviation of 1.94 indicate the spread of the values around the mean. A standard deviation of about 1.94 means that most values of $k$ lie within approximately 2 units from the mean. **Final answers:** - Mean ($p$) = 4.20 - Variance = 3.76 - Standard deviation $\approx$ 1.94